Questions: Use graphical differentiation to verify that d/dx ln x = 1/x. Graph the function y=ln x. Choose the correct graph below. Each graph is shown on [-1,10] by [-1,10].

Use graphical differentiation to verify that d/dx ln x = 1/x.

Graph the function y=ln x. Choose the correct graph below. Each graph is shown on [-1,10] by [-1,10].

Transcript text: Use graphical differentiation to verify that $\frac{d}{d x} \ln x=\frac{1}{x}$. Graph the function $y=\ln x$. Choose the correct graph below. Each graph is shown on $[-1,10]$ by $[-1,10]$.

Solution

Solution Steps

Step 1: Understanding the Problem

The problem requires us to use graphical differentiation to verify that the derivative of $ \ln x $ is $ \frac{1}{x} $. Additionally, we need to choose the correct graph of the function $ y = \ln x $ from the given options.

Step 2: Graphical Differentiation

Graphical differentiation involves analyzing the slope of the tangent line to the curve at various points. For the function $ y = \ln x $, the slope of the tangent line at any point $ x $ is given by the derivative $ \frac{d}{dx} \ln x = \frac{1}{x} $.

Step 3: Identifying the Correct Graph

The function $ y = \ln x $ is defined for $ x > 0 $ and has the following characteristics:

It passes through the point (1, 0) because $ \ln 1 = 0 $.
It increases slowly for large values of $ x $.
It has a vertical asymptote at $ x = 0 $.

By examining the given graphs:

Graph A shows a function that is undefined for $ x \leq 0 $ and has a vertical asymptote at $ x = 0 $, which matches the behavior of $ y = \ln x $.
Graph B and Graph C do not match the expected behavior of $ y = \ln x $.

Final Answer

The correct graph of the function $ y = \ln x $ is:

Option A

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