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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Differentiating Logarithmic Functions In Exercises 85–98, find the derivative of the function.

y = ln x ( 4 x 1 ) x 3 2

To determine

To calculate: The derivative of the function y=ln(x(4x1)x32).

Explanation

Given information:

The provided function is y=ln(x(4x1)x32).

Formula used:

The property of logarithms for the expression ln(xy) is ln(xy)=lnxlny where x and y are real numbers greater than 0.

The property of logarithms for the expression ln(xy) is ln(xy)=lnx+lny where x and y are real numbers greater than 0.

Constant multiple rule of derivative of function f(x) is f'(cx)=cf'(x) where c is constant.

The derivative of natural logarithm of a function is ddxlnu=1ududx, where u is a differentiable function of x.

The derivative of natural logarithm of x is ddxlnx=1x.

Calculation:

Consider the function, y=ln(x(4x1)x32).

Apply the property of ln(xy) as,

y=ln[x(4x1)]ln(x32)

Now, apply the property of ln(xy) as,

y=lnx+ln(4x1)ln(x32)

Then, the derivative of the function is,

y=ddx[lnx+ln(4x1)ln(x32)]=ddx(lnx)+ddx[ln(4x1)]ddx[ln(x32)]

Apply the chain rule where, u=4x1 as,

y=ddx(lnx)+1(4x1)ddx(4x1)ddx[ln(x32)]

Again apply the chain rule where, u=x32 as,

y=ddx(lnx)+1(4x1)ddx(4x1)1(x32)

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