Calculus I · 2A · lesson

The Derivative of the Natural Logarithm

Learn the derivative of the natural logarithm with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Course progress25 of 49 core pages25 of 49

Section overview

Trigonometric, exponential, and logarithmic functions

What this section is building

Learn the derivative of the natural logarithm with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Notice

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decide

Identify the function family first, then check whether a composition requires the chain rule too.

Avoid

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Concept

Learning objectives

Differentiate $\ln x$ and $\ln|x|$ ; connect the rule to inverse functions.

The Natural Logarithm

Explanation

Before the formulas

Special-function derivatives become less mysterious in The Derivative of the Natural Logarithm when you connect them to inverses and limits. The derivative of $\ln x$ follows from its inverse relationship with $e^x$ ; the derivative of tangent follows from sine, cosine, and the quotient rule. Learning these connections reduces memorization and provides recovery routes when a formula is forgotten.

Use a graph or a numerical point to check signs and scale. If the function is increasing rapidly, its derivative should be positive and large. If the function is undefined at an input, its derivative formula cannot rescue that input.

**Visual study**
The graphs of e x and ln x reflect across y=x. Tangent slopes at inverse points are reciprocals, which leads to (ln x)'=1/x.

Explanation

The logarithm measures how much exponent is needed

Because $\ln x$ reverses $e^x$ , its slope must undo the exponential slope. At the point $(x,\ln x)$ , the inverse point is $(\ln x,x)$ on $e^x$ , whose slope is $x$ . Reflection across $y=x$ turns that slope into its reciprocal $1/x$ .

The derivative becomes smaller as $x$ grows, matching the graph: equal multiplicative changes in $x$ produce equal additive changes in $\ln x$ , so large absolute changes matter less at larger scales.

The natural logarithm is the inverse of the natural exponential, but its derivative also has a direct geometric meaning: $1/x$ measures the relative effect of a fixed additive change at different scales. Adding one unit matters enormously near zero and barely matters when $x$ is huge.

This scale sensitivity is why logarithms appear in sound levels, information, growth comparisons, and elasticity. The domain restriction $x>0$ is part of the function, not a technical footnote to be remembered only after losing points.

Because $y=\ln x$ is the inverse of $x=e^y$ , its derivative is the reciprocal of the derivative of $e^y$ :

\boxed{\frac{d}{dx}\ln x=\frac1x},\qquad x>0.

More generally,

\boxed{\frac{d}{dx}\ln|x|=\frac1x},\qquad x\ne0.

Guided walkthrough

Differentiate a logarithmic product

Differentiate

f(x)=x^2\ln x, \qquad x>0.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A logarithm of an absolute value

For $f(x)=\ln|x-4|$ , the chain rule gives

f'(x)=\frac1{x-4},\qquad x\ne4.

The absolute value allows the logarithm to be defined on both sides of $4$ .

Interactive checkln-01

Differentiate $x^2\ln x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use the product rule and (ln x)'=1/x.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Application

Diminishing benefit from study time

Suppose a rough mastery score follows

M(t)=60+12\ln(1+t),

where $t$ is focused study time in hours. Then

M'(t)=\frac{12}{1+t}.

The model predicts large early gains and smaller later gains. At $t=1$ , the marginal gain is $6$ score units per hour; at $t=5$ , it is $2$ . This is diminishing return, not an argument to stop studying after one hour, however eagerly the algebra may be misquoted.

Optional advanced note

The logarithm as accumulated reciprocal growth

A later integration unit will define

\ln x=\int_1^x\frac{1}{t}\,dt.

The Fundamental Theorem of Calculus then gives $(\ln x)'=1/x$ immediately. This connects logarithms to area, multiplicative scaling, and the fact that equal percentage changes correspond to equal logarithmic differences.

Interactive checklog-extra-01

Differentiate $\ln(x^2+1)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use $u'/u$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

The Natural Logarithm

Explanation

Before the formulas

Read this graph as text

The logarithm inherits reciprocal slopes from its inverse. The graphs of e x and x reflect across y=x . Tangent slopes at inverse points are reciprocals, which leads to ( x)'=1/x . The point (1,e) on the exponential reflects to (e,1) on the logarithm. Reflection swaps horizontal and vertical change, so slopes become reciprocals. Since the exponential slope at input 1 is e , the logarithm slope at input e is 1/e . In general, the logarithm slope at x is 1/x .

The visual uses labeled positions, solid and dashed line styles, and written descriptions so the logarithm inherits reciprocal slopes from its inverse does not depend on color.

Why it matters: This visual links the logarithm derivative to the inverse-function rule and reinforces the geometric meaning of inverse functions. It should reduce the appearance that 1/x is an arbitrary entry in a formula table.

Visual study

The graphs of e x and ln x reflect across y=x. Tangent slopes at inverse points are reciprocals, which leads to (ln x)'=1/x.

Explanation

The logarithm measures how much exponent is needed

The natural logarithm is the inverse of the natural exponential, but its derivative also has a direct geometric meaning:

1/x

measures the relative effect of a fixed additive change at different scales. Adding one unit matters enormously near zero and barely matters when

x

is huge.

This scale sensitivity is why logarithms appear in sound levels, information, growth comparisons, and elasticity. The domain restriction

x>0

is part of the function, not a technical footnote to be remembered only after losing points.

Because

y=\ln x

is the inverse of

x=e^y

, its derivative is the reciprocal of the derivative of

e^y

\boxed{\frac{d}{dx}\ln x=\frac1x},\qquad x>0.

More generally,

\boxed{\frac{d}{dx}\ln|x|=\frac1x},\qquad x\ne0.

Guided walkthrough

Differentiate a logarithmic product

Differentiate

f(x)=x^2\ln x, \qquad x>0.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A logarithm of an absolute value

For $f(x)=\ln|x-4|$ , the chain rule gives

f'(x)=\frac1{x-4},\qquad x\ne4.

The absolute value allows the logarithm to be defined on both sides of $4$ .

Interactive checkln-01

Differentiate $x^2\ln x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use the product rule and (ln x)'=1/x.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Application

Diminishing benefit from study time

Suppose a rough mastery score follows

M(t)=60+12\ln(1+t),

where $t$ is focused study time in hours. Then

M'(t)=\frac{12}{1+t}.

Optional advanced note

The logarithm as accumulated reciprocal growth

A later integration unit will define

\ln x=\int_1^x\frac{1}{t}\,dt.

Interactive checklog-extra-01

Differentiate $\ln(x^2+1)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use $u'/u$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

The Natural Logarithm

Related resources

Original instruction with traceable references.

The Natural Logarithm

Related resources

Original instruction with traceable references.