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Problem 69 Medium Difficulty

Suppose $ f $ is differentiable on $ \mathbb{R}. $ Let $ F(x) = f(e^x) $ and $ G(x) = e^{f(x)}. $ Find expressions for
(a) $ F'(x) $ (b) $ G'(x). $

Answer

a. $=f^{\prime}\left(e^{x}\right) e^{x}$
b. $=e^{f(x)} f^{\prime}(x)$

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YL

Yang L.

March 31, 2021

(a) Find the differential dy. y = 1 x + 2

Video Transcript

here we have capital F of X, which is a composite function with little F as the outside function and eat of the X as the inside function. So it's derivative. Using the chain rule would be the derivative of the outside F prime of E to the X Times, the derivative of the inside E to the X. And then we have capital G of X, also a composite function. It's outside function is the each of the X function and it's inside. Function is f of X, so it's derivative would be the derivative of the outside e to the f of x times, the derivative of the inside F prime of X.