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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). $
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00:32
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 4
The Chain Rule
Derivatives
Differentiation
Yang L.
March 31, 2021
(a) Find the differential dy. y = 1 x + 2
Missouri State University
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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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.
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