Answered: In x In x + 1 31. y =

Math

Calculus

In x In x + 1 31. y =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 44RE

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

Derivatives Find the derivative of the following functions

Expert Solution

Step 1

Given function is $y = \frac{\ln x}{\ln x + 1}$

Quotient rule of derivative states that:

$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d}{d x} (u) - u \frac{d}{d x} (v)}{v^{2}}$

So, differentiating the given function with respect to x,

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{d}{d x} (\frac{\ln x}{\ln x + 1}) \\ = & \frac{(\ln x + 1) \frac{d}{d x} (\ln x) - \ln x \frac{d}{d x} (\ln x + 1)}{{(\ln x + 1)}^{2}} \\ = & \frac{(\ln x + 1) \cdot \frac{1}{x} - \ln x (\frac{1}{x} + 0)}{{(\ln x + 1)}^{2}} \\ = & \frac{\frac{1}{x} [(\ln x + 1) - \ln x]}{{(\ln x + 1)}^{2}} \end{array}$

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