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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the derivative of the function.

y = 1 + x e 2 x

To determine

To find: The derivative of the function y=1+xe2x.

Explanation

Given:

The function is y=1+xe2x.

Result used:

The Chain Rule:

If h is differntibale at x and g is differentiable at h(x), then the composite function F=gh defined by F(x)=g(h(x)) is differentiable at x and F is given by the product,

F(x)=g(h(x))h(x) (1)

Derivative Rule:

(1) Power Rule: ddx(xn)=nxn1.

(2) Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(1+xe2x)

Let h(x)=1+xe2x and g(u)=u  where u=h(x).

Apply the chain rule as shown in equation (1),

y(x)=g(h(x))h(x) (2)

The derivative g(h(x)) is computed as follows,

g(h(x))=g(u)=ddu(g(u))=ddu(u)

Apply the power rule (1) and then substitute u=1+xe2x,

g(h(x))=(12u121)=12u122=12u12=12(1+xe2x)12

Thus, the derivative is g(h(x))=12(1+xe2x)12

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