   Chapter 3.4, Problem 41E

Chapter
Section
Textbook Problem

Find the derivative of the function. f ( t ) = sin 2 ( e sin 2 t )

To determine

To find:  The derivative of the function f(t)=sin2(esin2t).

Explanation

Given:

The function is f(t)=sin2(esin2t).

Result used:

The Chain Rule:

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

F(t)=g(h(t))h(t) (1)

Calculation:

Obtain the derivative of f(t).

f(t)=ddt(f(t))=ddt(sin2(esin2t))

Let h(t)=esin2t and g(u)=sin2u  where u=h(t).

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

f(t)=g(h(t))h(t) (2)

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

g(h(t))=g(u)=ddu(g(u))=ddu(sin2u)=ddu(1cos2u2)   (Qsin2u=1cos2u2 )

Simplify further and obtain the derivative.

g(h(t))=ddu(12cos2u2)=ddu(12)ddu(cos2u2)=(0)(sin2u22)=sin2u

Substitute u=esin2t in the above equation,

g(h(t))=sin(2esin2t).

Thus, the derivative is g(h(t))=sin(2esin2t).

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

h(t)=ddt(esin2t)

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

Let k(t)=sin2t and s(u)=eu  where u=k(t).

h(t)=s(k(t))k(t) (3)

The derivative s(k(t)) is computed as follows,

s(k(t))=s(u)=ddu(s(u))=ddu(eu)=eu

Substitute u=sin2t in the above equation,

s(k(t))=esin2t

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