   Chapter 2.5, Problem 46E

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# Find the derivative of the function. y = [ x + ( x + sin 2 x ) 3 ] 4

To determine

To find:

The derivative of y=[x+x+sin2x)34

Explanation

Rule:

i. Power rule ddxxn=nxn-1

ii. Chain rule: dydx=dydu.dudx

iii. ddx(sinx)=cosx

iv. 2sinx.cosx=sin2x

Given:

y=[x+x+sin2x)34

Calculation:

Let y=u4

u=x+(x+sin2x) 3

dydx=ddx[x+x+sin2x)34

By using chain rule

dydx=dduu4.ddxu

By using power rule

dydx=4u3.ddxu

Substitute value of u

dydx=4[x+(x+sin2x) 3]3.ddxx+(x+sin2x) 3

By using sum rule

dydx=4[x+x+sin2x)33.ddx(x)+ddx (x+sin2x)3

By using power rule and power rule combined with chain rule

dydx=4[x+x+sin2x)33.1+3 (x+sin2x)2.ddx(x+sin2x

By using sum rule

dydx=4[x+x+sin2x)33

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