Exponential growth and logarithms (Properties of Logarithms) appear everywhere in science and finance. One special base makes their derivatives as simple as possible: the number $e \approx 2.71828$. The The Chain Rule then handles every other case.
🔑 Key idea
$$\frac{d}{dx}e^x = e^x, \qquad \frac{d}{dx}\ln x = \frac{1}{x}, \qquad \frac{d}{dx}a^x = a^x\ln a, \qquad \frac{d}{dx}\log_a x = \frac{1}{x\ln a}.$$ With an inside function: $\frac{d}{dx}e^{u} = e^{u}u'$ and $\frac{d}{dx}\ln u = \frac{u'}{u}$.
💡 Why it works
For $f(x) = a^x$: $\frac{a^{x+h} - a^x}{h} = a^x\cdot\frac{a^h - 1}{h}$. So the slope of $a^x$ is always $a^x$ times the slope at $x = 0$. The number $e$ is defined as the base whose slope at 0 is exactly 1, which makes $\frac{d}{dx}e^x = e^x$. For $\ln x$: since $e^{\ln x} = x$, differentiate both sides with the chain rule: $e^{\ln x}\cdot(\ln x)' = 1$, so $x\cdot(\ln x)' = 1$ and $(\ln x)' = \frac{1}{x}$. Finally, $a^x = e^{x\ln a}$, and the chain rule gives $a^x\ln a$.
For $y = e^x$, the slope at each point equals the height: the two graphs are identical.Equal steps in $x$ multiply $e^{rx}$ by the same factor, and the rate of growth is always proportional to the current size.
✏️ Worked example: Chain rule with $e$
Differentiate $e^{3x}$ and $e^{x^2}$.
$e^{3x}$: outer $e^{u}$ keeps its form; inner rate 3. So $3e^{3x}$.
$e^{x^2}$: inner rate $2x$. So $2xe^{x^2}$.
Answer: $3e^{3x}$ and $2xe^{x^2}$
✏️ Worked example: Logs
Differentiate $\ln(x^2 + 1)$ and $x\ln x$.
$\ln(x^2 + 1)$: derivative of the inside over the inside, $\frac{2x}{x^2 + 1}$.
$x\ln x$: product rule, $1\cdot\ln x + x\cdot\frac{1}{x} = \ln x + 1$.
Answer: $\frac{2x}{x^2 + 1}$ and $\ln x + 1$
✏️ Worked example: Other bases
Differentiate $5^x$ and $\log_2 x$.
$5^x\ln 5$.
$\log_2 x = \frac{\ln x}{\ln 2}$, so its derivative is $\frac{1}{x\ln 2}$.
Answer: $5^x\ln 5$ and $\frac{1}{x\ln 2}$
The rule $\frac{d}{dx}\ln u = \frac{u'}{u}$ reads as "derivative of the inside over the inside." It also gives $\frac{d}{dx}\ln\lvert x \rvert = \frac{1}{x}$ for negative $x$: there $\lvert x \rvert = -x$, and $\frac{-1}{-x} = \frac{1}{x}$. This fact will matter when we reverse it in Integrals Giving Exponentials and Logarithms.
The number $e$ also appears in compound interest: $\left(1 + \frac{1}{n}\right)^n \to e$ as $n \to \infty$. The two descriptions agree; we will see why with L'Hôpital's Rule.
⚠️ Common mistake
Writing $\frac{d}{dx}e^{3x} = e^{3x}$, forgetting the chain rule factor. $e^x$ is its own derivative, but $e^{3x}$ grows three times as fast. Check numerically: at $x = 0$, $\frac{e^{0.003} - 1}{0.001} \approx 3.0$, not 1.
Practice
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