Exercises

1. For each of the following functions h, (i) identify functions f and g so that h=f(g(x)). (ii) Find h'(x).
   1. .
   2. .
   3. .
2. Each of the following functions h is a nested composition of more than two functions. (i) identify all the functions that are used to make h. (ii) Find h'(x) by applying the chain rule two or more times.
   1. .
   2. .
   3. 
3. We can also use the chain rule to find the derivative of the function g if we know the function f. Let's consider two examples.
   1. We know that and are inverse functions, so that . By differentiating both sides of this equation (applying the chain rule to the left side), find a formula for the derivative of the natural logarithm function.
   2. We know that and are inverse functions, so that . By differentiating both sides of this equation (applying the chain rule to the left side), find a formula for the derivative of the inverse tangent function. (Hint: Your final answer should contain no trig functions. To obtain the final form of the derivative of , you will have to use facts from trigonometry.)
   3. Now let us consider the general case. Suppose that f and g are inverse functions, so that f(g(x)) =x. By differentiating both sides of this equation (applying the chain rule to the left side), find a formula for the derivative of g, the inverse function of f.