Chapter 9.7, Problem 30E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only. y = 3 x 4 x 4 + 3 3

To determine

To calculate: The simplified form of the derivative of y=3x4x4+33.

Explanation

Given Information:

The function is y=3x4xâˆ’33.

Formula used:

According to the power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

According to the product rule, if f(x)=u(x)â‹…v(x), then

fâ€²(x)=uâ€²(x)â‹…v(x)+vâ€²(x)â‹…u(x)

The derivative of a constant value, k, is

ddx(k)=0

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

y=3x4x4+33

Rewrite the function,

y=3x(4x4+3)13

Consider (4x4+3) to be u,

y=3xu13

Differentiate both sides with respect to x,

yâ€²=ddx(3xu13)=3ddx(xu13)

Simplify using the product rule,

yâ€²=3((ddx(x))â‹…u13+(ddx(u13))â‹…x)

Simplify using the power rule,

yâ€²=3((x1âˆ’1)â‹…u13+(13â‹…u13âˆ’1)â‹…x)=3(u13+(13uâˆ’23)â‹…x)

Take u13 common,

yâ€²=3u13(1+13uâˆ’23â‹…uâˆ’13â‹…ddx(u)â‹…x)=3u13(1+13uâˆ’1â‹…ddx(u)â‹…x

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