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Derivative of e^x — Definition, Formula & Examples

The derivative of e^x is simply e^x itself. This means the exponential function with base e is its own derivative — the rate of change of e^x at any point equals the function's value at that point.

If f(x)=exf(x) = e^x, then f′(x)=exf'(x) = e^x for all real xx. This result follows from the limit definition of the derivative and the fact that lim⁡h→0eh−1h=1\displaystyle\lim_{h \to 0} \frac{e^h - 1}{h} = 1, which is a defining property of the number ee.

Key Formula

ddx[ex]=ex\frac{d}{dx}\bigl[e^x\bigr] = e^x
Where:
  • ee = Euler's number, approximately 2.71828
  • xx = Any real number (the exponent)

How It Works

Because exe^x is its own derivative, differentiating it requires no extra computation — the answer is the same function you started with. When exe^x appears inside a composition, such as e3xe^{3x} or ex2e^{x^2}, you apply the chain rule: differentiate the outer e(⋅)e^{(\cdot)} (which stays as e(⋅)e^{(\cdot)}) and multiply by the derivative of the inner function. This self-replicating property is what makes exe^x central to solving differential equations, modeling continuous growth, and simplifying calculations throughout calculus and applied mathematics.

Worked Example

Problem: Find the derivative of f(x) = 5e^(3x).
Identify the structure: The function is a constant 5 multiplied by an exponential whose exponent is an inner function u = 3x.
f(x)=5 e3xf(x) = 5\,e^{3x}
Apply the chain rule: The derivative of e^u with respect to x is e^u multiplied by du/dx. Here du/dx = 3.
f′(x)=5⋅e3x⋅3f'(x) = 5 \cdot e^{3x} \cdot 3
Simplify: Multiply the constants together.
f′(x)=15 e3xf'(x) = 15\,e^{3x}
Answer: f′(x)=15e3xf'(x) = 15e^{3x}

Another Example

Problem: Prove that the derivative of e^x is e^x using the limit definition.
Write the limit definition: Start from the definition of the derivative.
ddx[ex]=lim⁡h→0ex+h−exh\frac{d}{dx}\bigl[e^x\bigr] = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h}
Factor out e^x: Use the exponent rule e^(x+h) = e^x · e^h to factor.
=lim⁡h→0ex(eh−1)h=exlim⁡h→0eh−1h= \lim_{h \to 0} \frac{e^x(e^h - 1)}{h} = e^x \lim_{h \to 0} \frac{e^h - 1}{h}
Evaluate the limit: The limit of (e^h − 1)/h as h → 0 equals 1. This is a foundational property of the number e.
=ex⋅1=ex= e^x \cdot 1 = e^x
Answer: ddx[ex]=ex\dfrac{d}{dx}\bigl[e^x\bigr] = e^x, confirmed from first principles.

Visualization

Why It Matters

This derivative rule appears constantly in AP Calculus AB/BC when solving differential equations like dydx=ky\frac{dy}{dx} = ky, whose solution is y=Cekxy = Ce^{kx}. In physics and engineering, exponential functions model radioactive decay, population growth, and electrical circuits — all situations where the rate of change is proportional to the current value. Mastering the derivative of exe^x and its chain-rule extension is essential before tackling integration techniques and series expansions.

Common Mistakes

Mistake: Forgetting the chain rule when differentiating e^(u(x)), writing the derivative as just e^(u(x)).
Correction: You must multiply by u'(x). For example, the derivative of e^(3x) is 3e^(3x), not e^(3x).
Mistake: Confusing the derivative of e^x with the power rule, writing xe^(x−1).
Correction: The power rule applies to x^n, not to e^x. The base here is the constant e, and the exponent is the variable. The derivative of e^x is e^x.

Related Terms