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(a) Use the Product Rule twice to prove that it $ f,g, $ and $ h $ are differentiable. then $ (fgh)' = f'gh + fgh'. + fgh'. $ (b) Taking $ f = g = h $ in part (a), show that

$ \frac {d}{dx}[f(x)]^3=3[f(x)]^2f'(x) $

(c) Use part (b) to differentiate $ y = e^{3x}. $

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01:19

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Email E.

February 26, 2022

consider the funtion R(x)

Harvey Mudd College

University of Michigan - Ann Arbor

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.

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(a) Use the Product Rule t…

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$\begin{array}{l}{\text { …

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The Product Rule can be ex…

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If $f, g$, and $h$ are dif…

01:51

Use the Product Rule twice…

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Verify that, if $f, g, h$ …

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Assume $f, g,$ and $h$ are…

he has clear. So when you right here. So for part Amy of you is equal to F G, then that makes the derivative of you beagle to the derivative of half times G plus tough times the derivative of gene. We've the product roll, so f g h this equal to you have a JJ. So when we derive this, this is equal to the derivative of U H, which is equal to the derivative of you terms H plus U turns the derivative of beach. So we got the derivative of F g H be equal to the derivative of afternoon's G plus F tongues, the derivative of G terms each plus G h Did they rip it if which is equal to the derivative of F tongues G H plus f times the derivative of G times H plus F G in the derivative of h, your part B, we're gonna make f B equal to G and be equal to age. And we have to formula in part a. And this gives us of cubed the derivative this equal to the first derivative of Times Square. Let's, uh, times the derivative of that times F plus F Square turns the derivative of which is equal to three terms, a derivative of of terms of square. In other words, do you over defects ah x You is equal to three times off Max square tongues the derivative after after backs. For part C, we're gonna use part B. So we have Why is equal to he to the three X, which is equal to e to the X cubed. This is the dirt formula for the derivative of a function cube. So we make up of X B equal to eat the X, and that makes the derivative. We also eat the axe the A D over D backs per eat the ex cute which is equal to D over t x f x cubed, which is people 23 turns e the X square turned Eat the ax when we go three me to the three x

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