Question

Use the Chain Rule to prove the following. a. The derivative of an even function is an odd function. b. The derivative of an odd function is an even function.

Answer 1

a. Let f(x) be an even function, then we have f(-x) = f(x) for all x in the domain of f. We want to show that the derivative of f(x), denoted as f'(x), is an odd function.

Using the chain rule, we have:

f'(-x) = (f(-x))' = (f(x))' = f'(x)

Now, let's evaluate f'(-x) + f'(x):

f'(-x) + f'(x) = f'(x) + f'(x) = 2f'(x)

Since f'(-x) + f'(x) simplifies to an expression that involves only f'(x), we can conclude that f'(x) is an even function.

b. Let g(x) be an odd function, then we have g(-x) = -g(x) for all x in the domain of g. We want to show that the derivative of g(x), denoted as g'(x), is an even function.

Using the chain rule, we have:

g'(-x) = (g(-x))' = (-g(x))' = -g'(x)

Now, let's evaluate g'(-x) + g'(x):

g'(-x) + g'(x) = -g'(x) + g'(x) = 0

Since g'(-x) + g'(x) simplifies to zero, we can conclude that g'(x) is an even function.

Explanation: