Lecture 8 · Barnett ch. 3 · about 40 min

Derivatives of Exponential and Log Functions

The two derivative rules that make exponential and logarithmic models tractable, combined with the chain rule, and the growth-rate reading of a logarithmic derivative.

By the end you can

Differentiate $e^x$, $\ln x$, and general bases $a^x$, $\log_a x$
Combine the exponential and log derivative rules with the chain rule
Differentiate a continuously compounded balance and interpret the result
Use the logarithmic derivative to read off a continuous growth rate

The derivative of $e^x$

The function $e^x$ has a remarkable property: it is its own derivative. This is really the defining property of $e$ , and it is why continuous growth models are built on it.

Derivative of the natural exponentialKey formulaFormula sheet →

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

$e$: Euler's number, \approx 2.71828

Use it when the base is e and the exponent is exactly x, with nothing more complicated in the exponent.

When the exponent is itself a function of $x$ , the chain rule is required, since $e^{g(x)}$ is a composition.

Chain rule with eFormula sheet →

\frac{d}{dx}\,e^{g(x)} = g'(x)\,e^{g(x)}

$g(x)$: any differentiable function in the exponent

Use it when the exponent is anything other than plain x, such as kx, x^2, or -0.03x.

The derivative of $\ln x$

Derivative of the natural logKey formulaFormula sheet →

\frac{d}{dx}\,\ln x = \frac{1}{x}

$x$: the input, restricted to x > 0

Use it when the argument is exactly x. This is also the rule behind $\int \frac{1}{x}dx = \ln|x| + C$ in later courses.

Chain rule with lnFormula sheet →

\frac{d}{dx}\,\ln\big(g(x)\big) = \frac{g'(x)}{g(x)}

$g(x)$: any differentiable, positive function

Use it when the argument of ln is anything other than plain x.

Bases other than $e$

General exponential and log derivativesFormula sheet →

\frac{d}{dx}\,a^x = a^x \ln a \qquad \frac{d}{dx}\,\log_a x = \frac{1}{x \ln a}

$a$: any positive base, a \neq 1

Use it when the base is a number other than e, such as 2^x or log base 10.

Both reduce to the $e$ and $\ln$ rules when $a = e$ , since $\ln e = 1$ .

Marginal revenue with an exponential demand curve

The logarithmic derivative and growth rates

Dividing $f'(x)$ by $f(x)$ gives a very useful quantity: the instantaneous percentage growth rate of $f$ . In fact, this ratio is exactly the derivative of $\ln(f(x))$ , by the chain rule.

Logarithmic derivativeGrowth rateFormula sheet →

\frac{d}{dx}\ln\big(f(x)\big) = \frac{f'(x)}{f(x)}

$f'(x)/f(x)$: the relative (percentage) rate of change of f at x

Use it when a question asks for a growth rate rather than an absolute rate of change — percent per period, not units per period.

Exam practice

Summary and review

$\dfrac{d}{dx}e^x = e^x$ ; with the chain rule, $\dfrac{d}{dx}e^{g(x)} = g'(x)e^{g(x)}$ .
$\dfrac{d}{dx}\ln x = \dfrac{1}{x}$ ; with the chain rule, $\dfrac{d}{dx}\ln(g(x)) = \dfrac{g'(x)}{g(x)}$ .
General bases: $\dfrac{d}{dx}a^x = a^x\ln a$ ; $\dfrac{d}{dx}\log_a x = \dfrac{1}{x\ln a}$ .
Exponential demand and revenue functions need the product rule and the chain rule together.
The logarithmic derivative $f'(x)/f(x)$ is the instantaneous relative growth rate; for $Ce^{kt}$ it is the constant $k$ .

The derivative of exe^xex

The derivative of ln⁡x\ln xlnx

Bases other than eee