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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 e x 1 + e x

To determine

To calculate: The derivative of the function y=ln(ex1+ex).

Explanation

Given information:

The provided function is y=ln(ex1+ex).

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 inverse property of logarithms for the expression lnex is lnex=x.

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

The derivative of e raised to x is ddxex=ex.

Calculation:

Consider the function, y=ln(ex1+ex).

Apply the property of ln(xy) as,

y=ln(ex)ln(1+ex)

Now, apply the inverse property of lnex as,

y=xln(1+ex)

Then, the derivative of the function is,

y=ddx[xln(1+ex)]=ddx(

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