5. Differentiate the following natural log and e functions.a.b.C.y' =y' =y'11y" =y = In(12 - x - 5x²1)y = In(e-x + xe-x)y = x² In (8x)

Sum rule of differentiationProduct rule of differentiationChain rule of differentiation

Answered: 5. Differentiate the following natural…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

5. Differentiate the following natural log and e functions.
a.
b.
C.
y' =
y' =
y'
11
y" =
y = In(12 - x - 5x²1)
y = In(e-x + xe-x)
y = x² In (8x)

Expert Solution

Step 1: Introduction

Sum rule of differentiation

$fraction numerator d left parenthesis u plus v right parenthesis over denominator d x end fraction equals fraction numerator d u over denominator d x end fraction plus fraction numerator d v over denominator d x end fraction$

Product rule of differentiation

$fraction numerator d left parenthesis u v right parenthesis over denominator d x end fraction equals v fraction numerator d u over denominator d x end fraction plus u fraction numerator d v over denominator d x end fraction$

Chain rule of differentiation

$fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d z end fraction cross times fraction numerator d z over denominator d x end fraction$

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