In this lesson, we're going to focus on differentiating inverse trigonometric functions. So let's get right into it. What is the derivative of arc sine of x cubed? Go ahead and try this problem.
Now you need to know the formula. If you know the formula, life is easy. The derivative of some generic arc sine function, let's say arc sine u, it's equal to u prime divided by the square root of 1 minus u squared.
So all you need to do is identify u and u prime. Well, the first part is straightforward because u is the stuff inside of the arc sine function. So we can see that u is equal to x cubed. Now u prime, the derivative of u, that's going to be the derivative of x cubed, which is 3x squared. So now we can get the answer.
So the derivative of arc sine x cubed is going to be going to be u prime which is 3x squared divided by the square root of 1 minus u squared or 1 minus x cubed squared so the final answer is 3x squared divided by the square root of 1 minus x to the sixth power and so this is the answer now let's try another example what is the derivative of art cosine 5x minus 9 go ahead and try that so here's the formula that you need the derivative of Arc cosine of U is very similar to the arc sine of u formula It's going to be negative u prime instead of positive u prime divided by the square root of 1 minus u squared So in this example, u is 5x minus 9. That's the stuff inside of the arc cosine function. And the derivative of 5x minus 9 is simply 5. So the answer is going to be negative 5 divided by the square root of 1 minus 5x minus 9 squared. And that's the solution for this problem. Here's another problem that you can work with.
What is the derivative of arctangent square root x? So if you don't have the formula, check out your textbook. It should be in it. But here it is for arctangent.
The derivative of arctan or inverse tan of u is going to be u prime divided by 1 plus u squared. So we can see that u is the square root of x, which if we rewrite it, that's x to the one half. Now u prime, that's, I brought u to the half for some reason, u prime is the derivative of x to the one half. So using the power rule, it's going to be one half x to the minus one half. And we can rewrite that as one over two x to the positive one half.
and then convert it back to radical form, it's 1 over 2 square root x. So now let's use the formula. It's u prime, which is square root x, divided by 1 plus, well actually, I take that back. u prime is this thing right here, so that's 1 over 2 square root x, and then divided by 1 plus u squared. So u is the square root of x.
And once we square it, it simply becomes x. So this is 1 over 2 square root x divided by 1 plus x. Now we could simplify the expression.
Now let's multiply the top and the bottom by 2 square root x. So we can get rid of the fraction in the numerator. So this is going to be 1 over 1 plus x times 2 square root x. And if you need to rationalize the expression, you can multiply the top and the bottom by the square root of x.
So on the denominator, the square root of x times the square root of x simply is x. So you can leave your final answer like that. So it's the square root of x divided by 2x.
times 1 plus x and so that's the inverse tangent or the derivative of our tangent of square root x. Let's consider one more example. Find the derivative of arc secant x to the fourth power. So let's begin with a formula.
So the derivative of arc secant is going to be u prime divided by the absolute value of u times the square root of u squared minus 1. So we can see that u is x to the 4th, and u prime is 4x cubed. So this is going to be u prime, which is 4x cubed, divided by the absolute value of u, which is x to the 4th. Now, because this term will always be positive, we really don't need the absolute value around it.
And then it's going to be the square root of u squared. So x to the 4th squared is really x to the 8th, and then minus 1. Now we could simplify this expression. We could cancel 3 of the 4x variables, leaving 1 in a denominator. So this is going to be 4, and this time I'm going to put back the absolute value symbol. So 4 over x times the square root of x to the 8th minus 1. And so that's the derivative of arc secant x to the 4th power.