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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the derivative of the function.

y = esin 2x + sin(e2x)

To determine

To find:  The derivative of the function y=esin2x+sin(e2x).

Explanation

Given:

The function is y=esin2x+sin(e2x).

Result used:

The Chain Rule:

If h is differentiable 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)

Sum Rule:

If f(x). and g(x) are both differentiable function, then

ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)] (2)

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(esin2x+sin(e2x))

Apply the sum rule as shown in equation (2),

y=ddx(esin2x+sin(e2x))

    =ddxesin2x+ddxsin(e2x) (3)

Obtain the derivative ddx(esin2x) using the chain rule as shown in equation (1),

Let h(x)=sin2x and f(u)=eu  where u=h(x)

ddx(esin2x)=g(h(x))h(x) (4)

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

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

Substitute u=sin2x in the above equation,

g(h(x))=esin2x

Thus, the derivative g(h(x)) is g(h(x))=esin2x .

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

h(x)=ddx(sin2x)=cos2x2=2cos2x

Thus, the derivative of h(x) is h(x)=2cos2x.

Substitute esin2x for g(h(x)) and 2cos2x for h(x) in equation (4),

g(h(x))h(x)=esin2x(2cos2x)ddx(esin2x)=2cos2xesin2x

Thus, the derivative is ddx(esin2x)=2cos2xesin2x

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