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

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00:59

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Boston College

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.

01:34

Suppose $f$ is differentia…

01:09

01:13

Let $f$ be a differentiabl…

01:22

Suppose $ f $ is different…

00:42

okay. And this problem capital F of X is f of X to the Alfa and we want to find f prime. So we're going to use the chain rule. The outside function is the F function. Lower case F. So it's derivative would be lower case f prime of X of out extra. The Alfa Times, the derivative of the inside the anti function is extra the Alfa and we can use the power rule on that Bring down the Alfa and then raise X to the Alfa minus one for G of X capital G of X is f of X to the Alfa Power. So we have similar functions but in a different order and we want g prime So again using the chain rule. Now the outside function is the Alfa power function. So we bring down the Alfa and we raise F of X to the Alfa minus one and now for the derivative of the inside, the derivative of F of X would be f prime of X

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