Derivatives of Exponential Functions (Introduction)

1 min read

Reference: Stewart §6.2

The Key Idea

Here’s a remarkable fact: the function $e^x$ is its own derivative.

$$\boxed{\frac{d}{dx}[e^x] = e^x}$$

No other function (except $f(x) = 0$) has this property. This makes $e^x$ incredibly important in calculus and its applications.

Quick Reference

Function	Derivative
$e^x$	$e^x$
$Ce^x$ (constant $C$)	$Ce^x$
$e^{kx}$	$ke^{kx}$

Why This Works

Recall that $e \approx 2.718$ is defined as the special base where: $$\lim_{h \to 0} \frac{e^h - 1}{h} = 1$$

When we differentiate $e^x$ using the limit definition: $$\frac{d}{dx}[e^x] = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} = e^x \cdot \lim_{h \to 0} \frac{e^h - 1}{h} = e^x \cdot 1 = e^x$$

Geometric Interpretation

At any point $(x, e^x)$ on the curve $y = e^x$, the slope of the tangent line equals the $y$-value at that point.

Point	$y$-value	Slope
$(0, 1)$	$1$	$1$
$(1, e)$	$\approx 2.72$	$\approx 2.72$
$(2, e^2)$	$\approx 7.39$	$\approx 7.39$

Constant Multiples

For any constant $C$: $$\frac{d}{dx}[Ce^x] = C \cdot \frac{d}{dx}[e^x] = Ce^x$$

Example: $\frac{d}{dx}[5e^x] = 5e^x$

Practice Problems

Level 1 Basic Derivatives

Find the derivative: (a) $f(x) = e^x$ (b) $g(x) = 7e^x$ (c) $h(x) = -e^x$

Thought Process

Use the fact that $\frac{d}{dx}[e^x] = e^x$. Constants multiply through.

Show Answer

(a) $f'(x) = \boxed{e^x}$

(b) $g'(x) = 7 \cdot e^x = \boxed{7e^x}$

(c) $h'(x) = -1 \cdot e^x = \boxed{-e^x}$

Level 2 Tangent Line

Find the equation of the tangent line to $y = e^x$ at the point where $x = 0$.

Thought Process

Find the point: $(0, e^0) = (0, 1)$. Find the slope: $f'(0) = e^0 = 1$. Use point-slope form.

Show Answer

At $x = 0$:

Point: $(0, e^0) = (0, 1)$
Slope: $f'(0) = e^0 = 1$

Tangent line: $y - 1 = 1(x - 0)$, so $\boxed{y = x + 1}$

Common Misconceptions

Common misconception

$\frac{d}{dx}[e^x] = xe^{x-1}$ because the power rule applies.

This is the concept-image-conflicts-definition error. The power rule $\frac{d}{dx}[x^n] = nx^{n-1}$ applies when a variable base is raised to a constant exponent. In $e^x$, the base is the constant $e$ and the exponent is the variable, so the power rule does not apply. The correct derivative is $e^x$, derived from the defining limit property of $e$.

Mastery Checklist

I know that $\frac{d}{dx}[e^x] = e^x$
I can find derivatives of constant multiples like $5e^x$
I understand why $e^x$ being its own derivative is special

What’s Next?

For more advanced techniques including:

The chain rule with exponentials: $\frac{d}{dx}[e^{u(x)}] = e^{u(x)} \cdot u'(x)$
General base exponentials: $\frac{d}{dx}[a^x] = a^x \ln a$
Product rule combinations

See the comprehensive page: Derivatives of Exponential Functions (MATH162)

Previous	Up	Next
Exponential Properties	Skills Index	Derivatives of Exponentials (Full)

Last updated: 2026-01-23