In this lesson, we're going to focus on finding the derivative of a function. So let's start with a constant. The derivative of any constant is equal to zero. So for instance, the derivative of the constant 5 is zero, and the derivative, let's say of negative 7, is also zero.
And you might be wondering, what exactly is a derivative? A derivative is a function that gives you the slope. at some x value.
So let's say if we have the function f of x is equal to 8. If we were to graph this function, it would look like a straight line at y equals 8. So around this region, let's say that's y equals 8. Now what is the slope of a straight line? The slope of a straight line is 0. And so if you were to find the derivative of this function, represented by f prime of x, that will give you 0. If you see d over dx it means that you're about to differentiate something with respect to x. So this means the derivative of f of x is equal to f prime of x. Now how can we find the derivative of a monomial?
For example, what is the derivative of x squared? Now there's something called the power rule. And the Powell rule is very useful for finding the derivative of monomials. So here's the formula that you want to use.
The derivative of a variable raised to a constant, such as x to the n, is equal to n times x raised to the n minus 1. And so that's the formula that you could use to find the derivative of a monomial. So in this case, n is equal to 2. So the derivative of x squared is going to be 2x to the 2 minus 1, which is 1, or basically 2 to the x power. So that's the derivative of x squared. Now let's try some other examples.
So using the same formula, that is the power rule, go ahead and find the derivative of these functions. So find the derivative of x cubed, x to the fourth, and also x to the fifth power. So the derivative of x cubed is going to be 3, in this case n is 3, so it's 3x raised to the 3 minus 1. And so 3 minus 1 is 2. So the answer is going to be 3x squared. Now, the derivative of x to the 4th power, in this case, n is 4, so it's going to be 4x raised to the 4 minus 1, and 4 minus 1 is 3, so it's 4x cubed. Now, for the last one, in this case, n is 5, so it's going to be 5x raised to the 5 minus 1, and 5 minus 1 is 4, so it's 5x to the 4th power.
And so that's a simple way in which you could find the derivative of a function. Now let's say if you want to find the derivative of 4x to the 7th power. How would you do it? How would you find the derivative of that particular monomial?
So what we need to do is use something called the constant multiple rule. And here it is. The derivative of a constant times a function, let's say a monomial, is going to be the constant times the derivative of that monomial or that function. Let's just put f of x here. So in this case, our c value is 4. And f of x...
is x to the seventh. So I'm just going to color code it. So c is 4, as you can see here, and f of x is x to the seventh. So let's bring out the 4 to the front, and then we're going to multiply it by the derivative of x to the seventh.
Now using the power rule, we can find the derivative of that function. So it's going to be 7x raised to the 7 minus 1. And so 7 minus 1 is 6. So we have 7x to the 6th power. And now...
we can multiply 4 and 7. 4 times 7 is 28. So the answer is going to be 28x to the 6th power. And so this is it. Now let's try some more examples.
So go ahead and find the derivatives of these two monomials. 8x to the 4th and also the derivative of 5x raised to the 6th power. So go ahead and take a minute.
So what we're going to do is move the constant to the front, and using the power rule, we're going to differentiate x to the 4th. So the derivative of x to the 4th is going to be 4 times x raised to the 4 minus 1, and 4 minus 1 is 3. And now we'll need to multiply 8 by 4. 8 times 4 is 32. So the answer is 32x. to the third power. Now for the next one, let's move the constant to the front, and then we're going to multiply by the derivative of x to the sixth power. So we can take this exponent, move it to the front.
So this is going to be 5 times the derivative of x to the sixth, which is 6x to the sixth minus 1, which is 5. So 6x to the 5th power. And now let's multiply 5 times 6. 5 times 6 is 30. So it's 30x to the 5th power. Now for the sake of practice, let's try a few more examples. So let's try the derivative of 9x to the 5th power.
And also the derivative of... 6x to the 7th power. So go ahead and try those two examples.
So this is going to be 9 times the derivative of x to the 5th power. And the derivative of x to the 5th power is 5x to the 4th power. And so 9 times 5, that's going to be 45. So the answer is 45x to the 4th power.
And so that's it for this one. Now for the last one, it's going to be 6 times the derivative of x to the 7th power. And the derivative of x to the 7th power is going to be, we can take the 7 and move it to the front. So that's going to be 7x to the 7 minus 1, which is 6. And 6 times 7 is 42. So it's 42x raised to the 6th power.
Now, we said that the derivative of x squared is equal to 2x. Now, how do we know that? By the way, let's say if f of x is x squared, that means that the derivative of f of x, which is f prime of x, is 2x. But how can we confirm this? Now, recall, the derivative is a function that can give you the slope.
at some x value. So we're going to show that soon. But first, is there another way in which we can get this answer besides using the power rule?
And in a typical calculus course, you need to know what that way is. And sometimes it's referred to as the definition of the derivative. Perhaps you've seen this function.
f prime of x is equal to the limit as h approaches zero of f of x plus h minus f of x divided by h. Now you might be wondering what exactly is f of x plus h? What is that?
Well we know what f of x is. f of x is x squared. So if f of x is x squared what is f of x plus h? All you need to do is replace the x with x plus h. So inside here we had x squared.
So instead of x, we're going to replace it with x plus h. So f of x plus h is x plus h squared. Now let's plug it into this formula.
So let me erase this. So remember, we're trying to show that f prime of x is equal to 2x. I'm going to erase that soon. Right now we have this f prime of x.
is equal to the limit as h approaches 0. And this right here, we know it's x plus h squared minus f of x, which is x squared, all divided by h. Now, what do you think we need to do at this point? What's our next step? Our next step is to FOIL that expression. So, this is equivalent to the limit as h approaches 0. And x plus h squared is the same as x plus h times x plus h.
So let's go ahead and FOIL that expression. Now, when taking a calculus exam, you will need to rewrite the limit expression. Even though it might be tedious, some teachers will actually take off points if you don't rewrite it. So here we have x times x, and that's going to be x squared, and then we have x times h, and then it's h times x, which is the same as x times h, and then the last one, h times h, so that's h squared, and then minus x squared divided by h. Now at this point, we can cancel the x squared term.
And we can combine like terms. XH plus XH, that's 2XH. So now we have 2XH plus H squared divided by H.
Now, our next step is to factor the GCF. That is the greatest common factor. which is h.
So if we take out an h from 2xh, that's going to be 2x. And h squared divided by h will give us h. So now we can cancel h. So now what we have left over is this. The limit as h approaches 0. of 2x plus h.
And so when h becomes 0, this is going to be 2x plus 0. So basically, h disappears as h approaches 0. And that's how we get the final answer, 2x. And so that's why the derivative of x squared is 2x. And so that's how you can find this answer using the limit process.
Now we said that the derivative is a function that will give you the slope at any x value. So let's say that f of x is x squared and we wish to find the slope of the tangent line at x equals 1. So we know what f prime of x is. Using the power rule, it's 2x.
And so to find the slope at x equals 1, we need to evaluate f prime of x when x is 1. And so that's going to be 2 times 1, which is 2. So the slope of the tangent line should be equal to 2. Now, if you were to draw a rough sketch of the graph y equals x squared, it will look something like this. And when x is equal to 1, the slope of the tangent line will equal 2. So I'm going to put... MT or MTAN.
So the slope of the tangent line is 2 when x is equal to 1. And so that's what the derivative function tells you. It gives you the slope of the tangent line at some x value. Now you need to know the difference between a tangent line and a secant line.
A secant line is basically a line that touches the curve at two points. And I missed it, so let's do that again. And a tangent line is a line that touches the curve.
only at one point. So make sure you know the difference between the two. Now in algebra, you've learned that to find the slope of a line, you need two points. And this is basically finding the slope of a secant line that's on the curve.
So let's put m secant. And you know that as y2 minus y1 equals x2 minus x1. So we could take two points on this curve. And Basically, get a secant line, and as those two points approach this point, the slope of the secant line approaches the slope of the tangent line. Now, we need to pick two points where the midpoint of those two points is x equals 1. So we can choose, let's say, 0.9 and 1.1 as our x1 and x2 values, because if you add up those two numbers and divide by 2, the average of 0.9 and 1.1 is 1. Or, we could pick 0.99 and 1.01, because the midpoint of those two numbers is still 1. However, 0.99 and 1.01 is closer to 1 than 0.9 and 1.1.
So the slope of the secant line, based on these two values, will be a lot closer to the slope of the tangent line at x equals 1. And so let's go ahead and calculate those values. So let's say that x1 is 0.9 to begin with, and x2 is 1.1. And let's use this formula to calculate the slope of the secant line. Now keep in mind the slope of the tangent line is this number.
It's equal to 2. So this is going to be y2 minus y1 divided by x2 minus x1. And so y2 corresponds to the y value for this x value. And y is equal to f of x.
So we could use this function f of x equals x squared to find y2. So when x2 is 1.1, y2 is 1.1 squared, because y equals x squared. When x1 is 0.9, y1 is 0.9 squared. Now, 1.1 squared, that's 1.21, and 0.9 squared is 0.81, and 1.1 minus 0.9 is 0.2. 1.21 minus 0.81 divided by 0.2 gives us already an exact answer, which is 2. And so there's no need to use 0.99 in this instance.
We can see that it's exactly the same. So let's try an example where it may not be exactly the same. So this time, let's say that f of x is x cubed. And we wish to calculate the slope of the tangent line at x equals 2. So we know what f prime of x is.
The derivative of x cubed using the power rule is 3x squared. So the derivative at x equals 2 is going to be 3 times 2 squared. 2 squared is 2 times 2, that's 4, times 3 is 12. So the slope... of the tangent line at x equals 2 is 12. Now let's see if we can approximate this value with the slope of the secant line. So let's choose an x1 value of 1.9 and an x2 value of 2.1.
And so the slope of the secant line between those two points is going to be y2 minus y1 over x2 minus x1. So in this case, we said x2 is 2.1. Now what's y2? y2 has to be 2.1 raised to the third power because y is equal to x cubed.
x1 is 1.9, so y1 is 1.9 to the third power. Now 2.1 raised to the third power, that's going to be 9. 261 and 1.9 raised to the third power that's going to be 6.859 and 2.1 minus 1.9 that's.2. 9.261 minus 6.859 that's 2.402 and if we divide that by.2 it gives us a very good approximation actually 12.01 and so You can see that you can approximate the slope of this tangent line using the slope of the secant line.
And that's what the derivative tells you. It gives you the slope of the tangent line which touches the curve at one point at some x value. In review, remember this. The derivative is a function that helps you to find the slope of a tangent line at some value of x.
So keep that in mind. Now let's talk about finding the derivative of a polynomial function. So let's say that f of x is x cubed plus 7x squared. minus 8x plus 6. What is the derivative of that function? So what is f prime of x?
So go ahead and work on this problem. So what we need to do is differentiate each monomial separately. Using the power rule, the derivative of x cubed is 3x squared.
Now what about the derivative of 7x squared? Using the constant multiple rule. it's going to be 7 times the derivative of x squared, which is 2x, or 2x to the first power. And 7 times 2x is 14x. Now, what about the derivative of negative 8x?
What is that equal to? So keep in mind, this is negative 8 times x to the first power. So this is going to be negative 8. times the derivative of x to the first power.
And what is the derivative of x to the first power? Well, using the power rule, we need to move the 1 to the front. So it's 1 times x raised to the 1 minus 1, which becomes negative 8 times 1x to the 0. Now, what is x to the 0? x to the 0, or anything raised to the 0 power, is 1. So this becomes negative 8 times 1, which is just negative 8. So the derivative of negative 8x is simply negative 8. And the derivative of any constant is 0. So we can stop it here. This is the answer.
f prime of x is 3x squared plus 14x minus 8. The video that you're currently watching is the first part of the entire video. For those of you who want access to the second part of the video, I'm going to put it on my Patreon page, as you can see the link in the screen. And on that page, I have some other video content that you might be interested in.
So feel free to take a look at that when you get a chance. Now let's get back to the lesson. Now let's say that f of x is 4. x to the fifth power plus 3x to the fourth power plus 9x minus 7. What is f prime of x?
So go ahead and try this for the sake of practice. So using the constant multiple rule, we're going to rewrite the constant and take the derivative of x to the fifth power using the power rule. And so that's going to be 5x to the fourth power. Now the derivative of x to the fourth is 4x cubed, and the derivative of x is always just 1, and the derivative of a constant is 0. And so we have 4 times 5, which is 20, and 3 times 4, which is 12. And so this is the final answer. f prime of x is 20x to the fourth power plus 12x cubed plus 9. Now let's say that f of x is 2x to the 5th power plus, let's say, 5x to the 3rd power plus 3x squared plus 4. And so you're given this function and you're told to find the slope of the tangent line at x equals 2. Go ahead and try this problem.
Anytime you need to find the slope of the tangent line, first you need to find the derivative of the function, that is f prime of x, and then simply plug in the x value into that function. So, let's determine f prime of x first. The derivative of x to the 5th power is 5x to the 4th power.
And the derivative of x cubed is 3x squared. And the derivative of x squared is 2x. And for the constant, we just don't need to worry about it. So f prime of x is going to be 10. 2 times 5 is 10. So 10x to the fourth power. 5 times 3 is 15. 3 times 2 is 6. And so this is what we have.
Now, to calculate the slope, let's replace x with 2. Now, 2 to the 4th power. If we multiply 2 four times, 2 times 2 is 4, times 2 is 8, times 2 is 16, and then 2 squared, that's 4, and 6 times 2 is 12. So now we have f prime of 2 is equal to 10 times 16, which is 160. 15 times 4, that's 60. And 160 plus 60, that's 220. And so the final answer is going to be 232. So that's the slope of the tangent line when x is equal to 2. Now, let's say that f of x is 1 over x. What is the derivative of that function? So what is the derivative of 1 over x?
How would you go about finding it? For a situation like this, you need to rewrite the function. And so what you need to do is take the x variable and move it to the top. When you do that, the exponent changes sign. It's going to change from positive 1 to negative 1. Now at this point, you could use the power rule.
So remember, the derivative of x to the n is n. x raised to the n minus 1. So in this case, n is negative 1, and negative 1 minus 1 is negative 2. So we have negative 1 x to the negative 2. Now once you have the derivative, you need to rewrite it into a more proper form. So let's take the x variable and move it back to the bottom. So our final answer will look like this.
It's negative 1 divided by x squared. And so that's how you could find it. the derivative of a rational function.
Now let's say that f of x is 1 over x squared. Go ahead and find f prime of x for the sake of practice. So take a minute and try that example. Just like before, we're going to rewrite the function.
So let's move the x variable to the numerator of the fraction. So f of x is equivalent to x raised to the negative 2. Now at this point, after you rewrite it, now you can find the derivative using the power rule. So let's take the exponent, move it to the front. So our n value is negative 2, and then let's subtract negative 2 by 1. Negative 2 minus 1 is negative 3. And now, let's rewrite this expression.
by taking the x variable and moving it to the bottom. If you're wondering why I divided by 1, it's the same thing. Negative 2x to the negative 3 is the same as negative 2x to the negative 3 over 1. I just like to write it in a fraction so you can see what I'm going to do next.
And that is moving the x variable to the bottom. So now the exponent will change from negative 3 to positive 3. And so this is the final answer. So this is the derivative of 1 over x squared.
Now let's try another example. So let's say that f of x, let's try a harder example. Let's say it's 8 over x to the 4th power. Go ahead and work on that problem. So let's rewrite the function.
Let's move the x variable. to the top and so this is going to be 8x raised to the negative fourth power and now let's differentiate the function. So we need to use the constant multiple rule in this case.
So it's going to be 8 times the derivative so I'm going to write that as times d over dx the derivative of x to the negative fourth power. And so, in this case, using the power rule, Our n value is negative 4. So it's going to be times negative 4x raised to the negative 4 minus 1. And so negative 4 minus 1 is negative 5. And 8 times negative 4 is negative 32. So the answer is negative 32x to the negative 5th power. But let's rewrite it though.
Let's not leave it like that. So if we move the x to the bottom, we can write the final answer fully simplified as negative 32 divided by x to the 5th power. Now let's talk about finding the derivative of radical functions. For instance, what is the derivative of the square root of x?
So what do you think we need to do? Now the first thing we need to do is rewrite this expression as a rational exponent. So x is the same as x to the first power, and if you don't see an index number, it's always a 2. So this is equivalent to x raised to the 1 half. Now in this form, we can use the power rule.
So n is going to be 1 half. So it's 1 half x raised to the 1 half minus 1. Now, we need to subtract a fraction by a whole number. And so we need to get common denominators.
1 is the same as 2 divided by 2. 2 divided by 2 is 1. And 1 over 2 minus 2 over 2, that's going to be negative 1 over 2. Now because we have a negative exponent, we need to rewrite this. So right now the 1 is on top and the x is also on top. The 2 is in the bottom of the fraction. But now I need to take the x variable and move it to the bottom. So this becomes 1 over 2 times x raised to the positive 1 half.
And at this point, I can rewrite the rational exponent as a radical. So we know that x to the 1 half is the square root of x. So the final answer is 1 divided by 2 square root x.
And so this is it. Now let's work on another example. So let's say that f of x is the cube root of x to the fifth power.
So what is f prime of x? Go ahead and try that. So let's begin by rewriting this expression.
So the cube root of x to the fifth power, we can rewrite that as a rational exponent, and it's going to look like this. It's x raised to the 5 over 3. So this number here becomes the numerator of the rational exponent and the index number becomes the denominator of the fraction that we see here. Now let's use the power rule.
So in this case, n is a fraction. It's going to be 5 over 3. And then we'll have x raised to the n minus 1. So that's 5 over 3 minus 1. Now, just like before, we need to get common denominators. 1 is the same as 3 divided by 3. And so we have 5 over 3 minus 3 over 3. 5 minus 3 is 2, so that becomes 2 over 3. Now, I'm going to rewrite this as 5 times x raised to 2 thirds divided by 3. Now, because the exponent is still positive, we don't need to move.
the x variable to the bottom and that's not necessary. The last thing that we need to do is convert this back into a radical expression. So this is going to be 5 times the cube root of x squared over 3. And so this is the final answer.
Here's another one that you can work on. So let's say that we have the monomial or rather just a radical expression, the seventh root of x to the fourth power. What is the derivative of that expression? So this is x raised to the 4 over 7, and let's use the power rule.
So n is 4 over 7, and it's going to be x to the 4 over 7 minus 1. To get common denominators, let's replace 1 with 7 over 7. Now, 4 over 7 minus 7 over 7, that's negative 3 over 7. So I'm going to rewrite this as a fraction. The 4 is on top, the x variable is currently on top, but the 7 is in the bottom of the fraction. Now, in this case, we do have a negative exponent, so we need to move the x variable to the bottom.
And so it's going to be 4 divided by 7 times x raised to the 3 over 7. And now we can rewrite the rational exponent as a radical expression. So the final answer is going to be 4 divided by 7 times the 7th root of x cubed. And that's it.
So this is the answer. Now of course, if you want to, you can rationalize the denominator, but I'm not going to worry about that in this video. Now, let's talk about some other problems that you might see in your homework.
So let's say if you're given a problem that looks like this. It has x squared on the outside, and then within a parenthesis it has x cubed plus 7. How would you find the derivative of this expression? What would you do? In this case, the best thing to do right now with what you already know is to distribute the x squared to x cubed plus 5. And then you can find the derivative.
So x squared times x cubed, that's going to be x to the fifth power because 2 plus 3 is 5. And then x squared times 7 is 7x squared. So in this form, it's very easy to find the first derivative. So the derivative of x to the 5th is 5x to the 4th.
And the derivative of x squared is 2x. So the final answer is 5x to the 4th power plus 14x. And so that's what you need to do if you ever come across a situation like that.
Now let's try a different example. Let's say that f of x is equal to 2x minus 3 raised to the second power. What would you do in this case?
Now, there's something called the chain rule, which we can use here, but you haven't learned that yet. So, we'll save that for another day, or rather, later on in this video. Something we can do is expand this expression. So whenever you see an exponent of 2, whatever that exponent is attached to, it means that you have two of these things multiplied to each other. So this expression is equivalent to 2x minus 3 times another 2x minus 3, which that does not look like a 3. And so what we're doing is we're multiplying a binomial by another binomial.
And so let's use the Fourier method. So let's multiply the first two terms. 2x times 2x. 2 times 2 is 4. x times x is x squared. And then 2x times negative 3. That's negative 6x.
Negative 3 times 2x is also negative 6x. And then we have negative 3 times negative 3, which is positive 9. Now, let's combine like terms. So, negative 6x plus negative 6x is negative 12x. And now we can find the first derivative. The derivative of x squared is 2x.
The derivative of x is 1. And for a constant, it's 0. So the final answer is going to be 8x minus 12. And so that's the derivative of 2x minus 3 squared. So that's what you could do in a situation like this. Now, let's say we have...
a fraction. x to the fifth plus 6x to the fourth power plus 5x cubed divided by x squared. In this case, what is the derivative of f of x? Now based on the previous examples, you know you need to simplify this before finding the derivative. So how can we simplify this expression?
If you're dividing a trinomial by a monomial, what you could do is divide every term by x squared separately. So let's begin by dividing x to the fifth by x squared. And it's important to understand that when you multiply, let's say x squared by x cubed, you need to add the exponents.
When you divide, you need to subtract. So this is five minus two, that's x cubed. So that's going to be the first part. So x to the fifth power divided by x squared is x cubed. Now 6x to the fourth power divided by x squared, that's going to be 6x squared.
All you need to do is subtract the exponents. 4 minus 2 is 2. And 5x cubed divided by x squared is going to be 5x to the first power. because 3 minus 2 is 1. And so now we have this simplified polynomial, and now we can find the first derivative. So it's going to be 3x squared, and the derivative of x squared is 2x, and the derivative of x is 1. So the final answer is 3x squared plus 12x plus 5. And that concludes this example.
Now let's talk about the derivatives of trigonometric functions. And I'm going to give you a few that you need to know. And for now, write these down because we're going to use this later.
Now the derivative of sine x you need to know is cosine x. And the derivative of cosine x is negative sine x. So that's the first two you need to know.
Next, you need to know that the derivative of secant x is secant x tangent x, and the derivative of cosecant x is actually very similar. It's going to be negative cosecant x. cotangent x.
Something that helps me to remember these things is that if you see a C in front you're going to have a negative sign like the derivative of cosine it was negative sign. Now consider the last two. The derivative of tangent is secant squared and based on that what do you think the derivative of cotangent x will be? Well notice that we do have the C. It turns out that it's negative cosecant squared x.
So keep those six derivative functions in mind because we're going to be using them later. Now the next thing that we're going to go over is the product rule. And here it is. So let's say if you have two functions multiplied to each other and you wish to find the derivative of that result. It's going to be the derivative of the first function times the second plus the first function times the derivative of the second.
So let's say if we wish to determine the derivative of x squared times sine x. So in this case, we could say that f is x squared and g is sine x. So I'm going to write it out. So if f is x squared. what is f prime?
f prime is the derivative of f and the derivative of x squared is 2x. Now g is going to be equivalent to sine x and g prime is the derivative of sine x which we now know is cosine x. So now at this point all we need to do is basically plug in what we have on the right side of the equation. So f prime is 2x g is sine x, f is x squared, and g prime is cosine x.
So this is the answer. If you want to, you can factor out the GCF, which is x, but I'm going to leave the answer like this. And so that's how you could use the product rule when finding the derivative of functions that are multiplied to each other. Now let's try some other examples.
Try this problem. What is the derivative of, let's say, 3x to the 4th power plus 7 times x cubed minus 5x? Now granted, we can FOIL this expression because we did an example like that earlier, but let's use the product rule to get the answer.
Feel free to pause the video if you want to. So what I'm going to do first is I'm going to write the formula. So the derivative of f times g is going to be the derivative of the first part times the second plus the first part, let me write that again, times the derivative of the second.
So what's f and what's g? We're going to say that f is the first part. And so we're going to say that f is 3x to the fourth power plus 7. So what is f prime?
So the derivative of x to the fourth is 4x cubed, but we're going to multiply that by 3. And so 3 times 4, that's going to give us 12. So this is going to be 12x cubed, and the derivative of 7 is 0. Now g is the second part of the function, so g is x cubed minus 5x. g prime is going to be... 3x squared and the derivative of x is 1 times negative 5 so that's just going to be negative 5 and so that's g prime.
So using the formula this expression becomes equal to which I'm going to write over here it's going to be f prime which is 12x cubed times g which is x cubed minus 5x plus f And that's 3x to the 4th plus 7 times g prime. 3x squared minus 5. And that's it right there. Now, here is a challenge problem for you. What is the derivative of x cubed times tangent x times 3x squared minus 9? So this time we have three parts being multiplied to each other.
So we saw how to use the product rule when having two different functions being multiplied to each other. But what about three different functions? So if the derivative of, let's say, a two-part function like f times g, if that's f prime g plus f g prime, what would the derivative of, let's say, F times G times HB.
So this is going to be, we're going to differentiate the first part, and then leave the second two parts the same, plus we're going to leave the first part the same, differentiate the second part, and then leave the third part the same, and then it's going to be the first two parts times the derivative of the last part. So when using the product rule, when you differentiate one part, the other two parts should remain the same. And then you just go in order. Differentiate the first part, and then the second part, and then the third part.
So once you understand the format, or the procedure of doing this, you can just go ahead and get the answer without actually writing down what's f, g, and h. So first, let's find the derivative of the first part. The derivative of x cubed is 3x squared. Now the other two parts... The g and h, we're just going to rewrite it for now.
So it's going to be times tangent x and then times 3x squared minus 9. Now, let's rewrite the first part, which is x cubed. And then we're going to take the derivative of the second part. The derivative of tangent, if you remember, is secant squared.
Now, let's rewrite the third part, which is 3x squared minus 9. Now for the last part, we're going to rewrite the first two parts, x cubed and tangent x, but this time we are going to take the derivative of 3x squared minus 9, which is going to be just 6x, because this will go to 0. Now, let's say if we want to find the derivative of a fraction. such as let's say 5x plus 6 divided by 3x minus 7. In this case, you want to use something called the quotient rule. And here's the formula that we're going to use. So the derivative of let's say f divided by g, this is going to be g f prime minus f g prime divided by g squared. And this is something that you simply need to commit to memory.
You just got to know that function. At least it worked for me when I was in high school. I just memorized that function. Now, f is going to be the top portion of this function. So in this case, f is going to be 5x plus 6. f prime is the derivative of f, so the derivative of 5x plus 6 is 5. Now, g is going to be the bottom part of this function.
So g is going to be 3x minus 7. which means G prime is 3. So if it helps to write everything out, by all means go ahead and do that. If it makes your life easier or if it helps you to avoid mistakes. And on a test, one of the biggest things that you have to do is avoid mistakes.
Because if you make a mistake even if you know it, I mean that's just going to ruin your test score. Now let's go ahead and finish this. G is 3x minus 7 and then f prime that's 5 and then we have f which is 5x plus 6 and then g prime that's 3 and then divided by g squared so g is 3x minus 7 and then let's square that.
Now in this case I'm going to simplify because It doesn't require that much work to do so. So let's begin by distributing the 5 to 3x minus 7. So 5 times 3x, that's 15x. And then 5 times 7, or 5 times negative 7 rather, that's negative 35. And here we have negative 5x times 3, which is going to be negative 15x. And 6 times 3 is 18, but we got the negative sign, so that's going to be negative 18. And I'm not going to FOIL the stuff on the bottom because it looks better this way.
Now we could cancel 15x. Negative 35 minus 18, that's going to be negative 53. And so the answer is negative 53 divided by 3x minus 7 squared. If it's easy to simplify your answer, feel free. But sometimes, if it takes a lot of work to simplify it, Most teachers will allow you just to write the answer the way it is. Some teachers will allow you to leave the answer like this, so you need to basically know your teacher and how they want you to write the final answer.