Chapter 9.6, Problem 14E

### 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

# Differentiate the functions in Problems 3-20. y = 1 ( 3 x 3 + 4 x 1 ) 3 / 2

To determine

To calculate: The derivative of the function y=1(3x3+4x+1)3/2.

Explanation

Given Information:

The provided function is y=1(3x3+4x+1)3/2.

Formula used:

Power rule for a real number n is such that, if y=un then dydx=nunâˆ’1â‹…dudx, where u is a differentiable function of x.

Power of x rule for function f(x)=xn is fâ€²(x)=nxnâˆ’1, where n is a real number.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

If any nonzero real number a has a negative integer n as its exponent then aâˆ’n=1an.

Calculation:

Consider the function,

y=1(3x3+4x+1)3/2

Rewrite the function as,

y=(3x3+4x+1)âˆ’3/2

Consider (3x3+4x+1) to be u,

y=uâˆ’3/2

Differentiate both sides with respect to x,

yâ€²=ddx(uâˆ’3/2)

Use the power rule,

yâ€²=âˆ’32â‹…uâˆ’32âˆ’1â‹…dudx=âˆ’32uâˆ’5/2dudx

Substitute (3x3+

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