Derivatives of Exponential and Logarithmic Functions, calculus lesson

A population that grows by a fixed percentage each year grows faster when it is larger. In other words, its rate of change is proportional to its size. Exponential functions are exactly the functions with that property, and one base, the number $e$ , makes the constant of proportionality equal to 1. Logarithms, as their inverses, then get derivatives that are just as simple.

Key terms

The natural exponential function

Definition The number $e \approx 2.71828$ is the unique base $a > 0$ for which

\lim_{h \to 0} \frac{a^h - 1}{h} = 1.

For this base, the function $e^x$ is differentiable at every real $x$ and

\frac{d}{dx} e^x = e^x,

because as $h \to 0$

\frac{e^{x + h} - e^x}{h} = e^x \cdot \frac{e^h - 1}{h} \to e^x \cdot 1 = e^x.

In plain words At every point the slope of $y = e^x$ equals its height. At $x = 0$ the graph crosses the $y$ axis with slope exactly 1.

Example The tangent line to $y = e^x$ at $x = 0$ is $y = 1 + x$ . At $x = 0.1$ the line gives $1.1$ , while $e^{0.1} \approx 1.10517$ . For $f(x) = x e^x$ the product rule gives $f'(x) = e^x + x e^x = (1 + x)e^x$ .

Exponentials with base a

Definition For a constant $a > 0$ , the exponential with base $a$ is defined for every real $x$ by $a^x = e^{x \ln a}$ . The function $a^x$ is differentiable on $\mathbb{R}$ and

\frac{d}{dx} a^x = a^x \ln a.

This is the chain rule applied to $e^{x \ln a}$ .

In plain words Every exponential has a slope proportional to its height. The constant is $\ln a$ , which is positive when $a > 1$ , zero when $a = 1$ and negative when $0 < a < 1$ .

Example For $f(x) = 2^x$ the slope at $3$ is $f'(3) = 2^3 \ln 2 = 8 \ln 2 \approx 5.545$ . For $g(x) = \left(\frac{1}{2}\right)^x$ the constant is $\ln \frac{1}{2} = -\ln 2$ , so

g'(x) = -\left(\frac{1}{2}\right)^x \ln 2 < 0

for every $x$ , and $g$ is decreasing.

Natural logarithm

Definition The natural logarithm $\ln x$ is the inverse of $e^x$ . It is defined for $x > 0$ , with $e^{\ln x} = x$ for $x > 0$ and $\ln(e^x) = x$ for all real $x$ . It is differentiable on $(0, \infty)$ and

\frac{d}{dx} \ln x = \frac{1}{x}, \qquad x > 0.

Since $\ln$ is differentiable (it is the inverse of a function with nonzero derivative), differentiating $e^{\ln x} = x$ by the chain rule gives $e^{\ln x} \cdot (\ln x)' = 1$ , so $(\ln x)' = \frac{1}{x}$ .

In plain words The logarithm always increases, but more and more slowly. At $x = 100$ its slope is only $0.01$ .

Example For $f(x) = x \ln x$ with $x > 0$ the product rule gives

f'(x) = \ln x + x \cdot \frac{1}{x} = \ln x + 1.

This is zero at $\ln x = -1$ , that is at $x = \frac{1}{e} \approx 0.368$ , where the graph has a horizontal tangent.

Logarithms with other bases

Definition For a constant $a > 0$ with $a \neq 1$ , $\log_a x$ is the inverse of $a^x$ , defined for $x > 0$ . The change of base formula

\log_a x = \frac{\ln x}{\ln a}

shows that it is differentiable on $(0, \infty)$ with

\frac{d}{dx} \log_a x = \frac{1}{x \ln a}, \qquad x > 0.

The base $a = 1$ is excluded because $1^x = 1$ has no inverse, and $\ln 1 = 0$ would sit in the denominator.

In plain words Every logarithm is a constant multiple of $\ln x$ , so its derivative is the same $\frac{1}{x}$ scaled by $\frac{1}{\ln a}$ .

Example For $\log_{10} x$ at $x = 10$ the slope is $\frac{1}{10 \ln 10} \approx 0.04343$ .

Logarithm of an absolute value

Definition The function $\ln \lvert x \rvert$ is defined for every real $x \neq 0$ , and on that set

\frac{d}{dx} \ln \lvert x \rvert = \frac{1}{x}.

For $x > 0$ this is the rule for $\ln x$ . For $x < 0$ we have $\ln \lvert x \rvert = \ln(-x)$ , and the chain rule gives

\frac{d}{dx} \ln(-x) = \frac{1}{-x} \cdot (-1) = \frac{1}{x}.

In plain words The formula $\frac{1}{x}$ works on both sides of zero once you put absolute value bars inside the logarithm. This is the version you need when $x$ can be negative.

Example At $x = -2$ the graph of $y = \ln \lvert x \rvert$ has slope $\frac{1}{-2} = -\frac{1}{2}$ . The graph is a mirror image of $y = \ln x$ , so a negative slope on the left matches the positive slope $\frac{1}{2}$ at $x = 2$ .

Derivative formulas and domains

Definition The basic formulas, each valid exactly on the domain shown, are collected here. Combined with the chain rule, for a differentiable function $u$ of $x$ and a constant $a > 0$ , they give

\frac{d}{dx} e^{u} = e^{u} u', \qquad \frac{d}{dx} a^{u} = a^{u} \ln a \cdot u', \qquad \frac{d}{dx} \ln \lvert u \rvert = \frac{u'}{u},

where the last formula holds wherever $u \neq 0$ .

Function	Derivative	Valid for
$e^x$	$e^x$	all real $x$
$a^x$ , $a > 0$	$a^x \ln a$	all real $x$
$\ln x$	$\frac{1}{x}$	$x > 0$
$\log_a x$ , $a > 0$ , $a \neq 1$	$\frac{1}{x \ln a}$	$x > 0$
$\ln \lvert x \rvert$	$\frac{1}{x}$	$x \neq 0$

In plain words Use a formula only where both the function and the formula make sense. Simplifying a logarithm can quietly change the domain.

Example The functions $f(x) = \ln(x^2)$ and $g(x) = 2 \ln x$ look the same, but $f$ is defined for all $x \neq 0$ and $g$ only for $x > 0$ . On its domain

f'(x) = \frac{2x}{x^2} = \frac{2}{x}.

Similarly the formula

\frac{d}{dx} \ln(x^2 + 1) = \frac{2x}{x^2 + 1}

holds for every real $x$ , since $x^2 + 1 > 0$ always.

Check yourself

Differentiate $f(x) = 5^x$ and find $f'(0)$ .

Show the answer

The derivative is $f'(x) = 5^x \ln 5$ , so $f'(0) = \ln 5 \approx 1.609$ .

Differentiate $y = \ln(\cos x)$ and say where your answer holds.

Show the answer

By the chain rule

y' = \frac{-\sin x}{\cos x} = -\tan x.

This holds wherever $\cos x > 0$ , for example on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , since $\ln$ needs a positive input.

Find the slope of $y = e^{3x}$ at $x = 0$ .

Show the answer

By the chain rule $y' = 3e^{3x}$ , so the slope at $0$ is $3$ .

Derivatives of Exponential and Logarithmic Functions

Theory, part 53

Practice, part 54

Key terms

The natural exponential function

Exponentials with base a

Natural logarithm

Logarithms with other bases

Logarithm of an absolute value

Derivative formulas and domains

Check yourself

How this lesson connects

Builds on

Used later in