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Problem

Suppose $ f $ is differentiable on $ \mathbb{R}. …
00:42

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Problem 68 Hard Difficulty

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

Related Courses

Calculus 1 / AB
Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Related Topics

Derivatives

Differentiation

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04:40
Derivatives - Intro

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.
Video Thumbnail

44:57
Differentiation Rules - Overview

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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Video Transcript

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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Related Topics

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Video Thumbnail

04:40
Derivatives - Intro

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.
Video Thumbnail

44:57
Differentiation Rules - Overview

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.
Join Course
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