Question

Derive a formula for the second derivative f(g(x)).

A) Use the power rule
B) Apply the chain rule
C) Substitute values
D) Ignore differentiation

Answer 1

The formula for finding the second derivative of f(g(x)), using the chain rule, is: f''(g(x)) = f''(g(x)) * g'(x)^2 + f'(g(x)) * g''(x).

Step-by-step explanation:

The formula for finding the second derivative of f(g(x)), using the chain rule, is:

f''(g(x)) = f''(g(x)) * g'(x)^2 + f'(g(x)) * g''(x)

Here's a step-by-step explanation:

1. Take the derivative of the outer function f(g(x)), which gives you f'(g(x)).
2. Take the derivative of the inner function g(x), which gives you g'(x).
3. Square g'(x) to get g'(x)^2.
4. Take the derivative of f'(g(x)) with respect to x, which gives you f''(g(x)) * g'(x).
5. Take the derivative of g''(x) with respect to x.
6. Multiply the result from step 3 with the result from step 4 and the result from step 5.