Derivatives of inverse functions Goal 1: You will learn the derivatives for exponential functions. _ a We will use this fact: E09") 2 9" The proof of this derivative is not super intuitive, so we are just going to use this derivative without proving it. 1. Find dx (2 ). 0 Some facts to help you do this: i. Recall that ex and ln(x) are inverses of each other. That means that em\") = x and ln(e") = x. In this case, we can see that 2 = 9W2). ii. (3(a)b 2 Jr\" iii. You can use the chain rule to differentiate a function ofthe form .99\") Goal 2: You will use implicit differentiation to help you find the derivative of several common inverse functions. You will also do some work with right triangles and trig functions. First, practice using implicit differentiation: 1. Use implicit differentiation to find ::, given that 4165 + tan(y) = y2 + 5x. Next, you are going to find - (In(x)). Again these pieces of information will be helpful: . Recall that ex and In(x) are inverses of each other. That means that eln(x) = x and In(ex) = x. The derivative of ex is ex. That is, my (ex) = ex. 2. Let y = In(x). Find a by taking the following steps: a. Solve the equation y = In(x) for x by applying the idea of inverse functions. b. Use implicit differentiation to find ax- c. Rewrite your expression for ~ so that it is expressed in x. (That is, your formula for a shouldn't contain any y's). 3. Now you are going to find - (arcsin(x)) by following the same steps as above. a. Solve the equation y = arcsin(x) for x by applying the idea of inverse functions.b. Use implicit differentiation to find . c. Rewrite your expression for a so that it is expressed in x. (That is, your formula for " shouldn't contain any y's. I suggest you use a right triangle to help you do this, and remember that sine opposite hypotenuse and cosine = adjacent hypotenuse 4. Follow the same steps as above to find ax (arctan(x))