in this lesson we're going to focus on simplifying derivatives so let's start with an example problem what is the derivative of this function x cubed times 2x minus 5 raised to the fourth power now for this problem we need to use the product rule and here's the formula for that so the derivative of f times g is the derivative of the first part f prime times the second plus the first part times the derivative of the second part so we can clearly see that f is x cubed and g is 2x minus 5 raised to the fourth power now f prime the derivative of x cubed that's going to be 3x squared and to find g prime we need to use the chain rule so the derivative of two x minus five raised to the fourth power is gonna be four and then keep the inside the same two x minus five using a power rule subtract this by one and then differentiate the inside function 2x minus 5 which is 2. so now let's apply this so it's going to be f prime which is 3x squared times g which is 2x minus 5 raised to the fourth power plus f that's x cubed and then g prime which is four times two and that's eight so it's going to be times eight two x minus five cubed now what should we do once we get to this part how can we simplify this expression the best thing to do right now is to take out the gcf the greatest common factor and so we can factor out an x squared and we could take out two x minus five at least three of them because there's three here and four here so we gotta take out the lower of the two numbers so if we take out x squared and two x minus 5 to the third power from that expression what's left over so the x squared is gone we took out three of these so there's gonna be one left over with a three in front so it's going to be 3 times 2x minus 5. now for the next one we took out x squared so there's an x left over we have an 8 and we take out all three of 2x minus 5's so it's going to be plus 8 x and that's all that's left over for that term now all we need to do is distribute the three and combine like terms so three times two x that's six x three times negative five that's negative 15 plus 8x and now let's add these two so we're going to have x squared times 2x minus 5 to the third power and 6x plus 8x that's 14x so this is the final answer fully simplified now let's work on another example feel free to pause the video if you want to try it what is the derivative of x squared times the square root of 4 minus 9x so once again we need to use the product rule and so you could rewind it if you need to look at the formula but the gist of it is that you need to differentiate the first part of the function which is x squared so that's 2x and then keep the second part the same so we're going to rewrite the square root of 4 minus 9x plus keep the first part the same and then differentiate the second part now let's focus on that because to differentiate this we need to use the chain rule and before we do that we need to rewrite it as four minus nine x raised to the one half now whenever you're differentiating a composite function let's say if you want to find the derivative of f of g of x based on the chain rule you need to differentiate the outside part of f and then keep the inside part the inside function g of x the same and then multiply by the derivative of the inside function so that's the main idea behind the chain rule so first let's differentiate the outside part of the function by using the power rule so it's going to be one half and then keep the inside function g of x the same so four minus nine x and then subtract the exponent by one so one half minus one or one half minus two over two that's negative one over two and then multiply by g prime of x the derivative of the inside function so the derivative of four minus nine x that's negative nine so that's what we have now so i'm going to combine these two so that's gonna be times negative nine over two and then four minus nine x raised to the negative one half now i'm going to rewrite this expression so let's erase it first and let's write it as a rational exponent so it's four minus nine x to the one half now in this form this is when you want to factor the expression where you want to take out the greatest common factor so we could take out an x from both terms because they both contain an x now in the last example we had 2x minus 5 raised to the fourth power and the first term and in the third power for the second and when you're dealing with four and three you want to take out the lowest of the two numbers so you take out three now in this case we have a negative exponent so what should we factor out one-half or negative one-half well negative one-half is lower than one half on the number line so because it's less we need to factor out four minus nine x to the negative one half now when you take out the gcf if you want to find out what goes on the inside divide what you had by the gcf so if we divide 2x by x the x variables will cancel and we're going to get 2. if we divide these two we need to subtract the exponents for instance let's say if you want to divide x to 7 by x to 4. this is going to be x to the 7 minus 4 which is x cubed so in this case if we wish to divide these two we're going to take one half and subtract it by negative one-half which becomes one-half plus one-half which is a whole so this is going to be four minus nine x raised to the first power which is great now let's focus on the second term negative nine over two x squared divided by x it's going to be negative nine over two times x and then if we divide these two negative one-half minus negative one-half that's gonna be zero so this is completely gone we took it out so this is what we have now so first let's distribute 2. so this is going to be 2 times 4 which is 8 and then 2 times negative 9x that's negative 18x and then minus 9 over 2x now we need to get common denominators so 18x i'm going to multiply by 2 over 2. so it's going to be negative 36 x over two and then minus nine over two x so negative thirty six minus nine is negative forty five so we have eight minus forty 45 over two x and then that's multiplied by x times four minus nine x raised to negative one half what we can do at this point is take this term or that factor and move it to the bottom to make the negative exponent positive so this is going to give us x times 8 minus 45 over 2 x divided by 4 minus 9x to the positive one-half now we can write the final answer as x times this stuff and then let's convert this rational exponent into a radical so that's simply the square root of 4 minus 9x so you can leave your answer like this if you want to but if you're told to rationalize the denominator you can take it one step further so if you need to do that you can multiply the top and the bottom by the square root of four minus nine x so you can also leave your answer as x 8 minus 45 x over 2 times the square root of 4 minus 9x divided by 4 minus 9x so if you need to rationalize it you can report your answer in this form but if it's not necessary to rationalize it this form is also acceptable there's a lot less writing in this form you