   Chapter 3.2, Problem 20E

Chapter
Section
Textbook Problem

Differentiate. y = ( z 2 + e z ) z

To determine

To find: The differentiation of the function y=(z2+ez)z.

Explanation

Derivative rule:

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Natural exponential function: ddx(ex)=ex

Calculation:

The derivative of the function y=(z2+ez)z is dydz, which is obtained as follows,

dydz=ddz((z2+ez)z)=ddz((z2+ez)z12)

Use product rule (1) and simplify the terms,

dydz=(z2+ez)ddz(z12)+(z12)ddz(z2+ez)

Apply the derivative rule (3), (2) and (4),

dydz=[(z2+ez)ddz(z12)]+[(z12)[ddz(z2)+ddz(ez</

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