AP Calculus AB and BC

Derivative of e^(-x): Answer, Chain Rule, Mistakes

The derivative of e to the negative x with respect to x is negative e to the negative x. The chain rule multiplies the exponential by the derivative of the exponent negative x, which is negative 1, so the result equals -e^(-x). The function is its own derivative up to that minus sign.

ddx[e−x]=−e−x\frac{d}{dx}\left[e^{-x}\right] = -e^{-x}

How to differentiate e^(-x)

The outer function is the exponential and the inner function is the exponent −x-x. The chain rule differentiates the outer function, keeps the inner one, then multiplies by the inner derivative.

ddxeu=eu⋅dudx\frac{d}{dx}e^{u} = e^{u}\cdot\frac{du}{dx}

With u=−xu = -x, the exponential reproduces itself and dudx=−1\frac{du}{dx} = -1.

ddxe−x=e−x⋅(−1)=−e−x\frac{d}{dx}e^{-x} = e^{-x}\cdot(-1) = -e^{-x}

What the sign of the derivative tells you

Since e−xe^{-x} is always positive, its derivative −e−x-e^{-x} is always negative. The function decreases everywhere, matching its graph: a curve that falls from left to right and approaches 00 as x→∞x \to \infty.

Each derivative flips the sign again, so d2dx2e−x=e−x\frac{d^{2}}{dx^{2}}e^{-x} = e^{-x} and the pattern alternates e−x,−e−x,e−x,…e^{-x}, -e^{-x}, e^{-x}, \dots. This is why e−xe^{-x} is the backbone of the exponential decay models in Unit 7.

The reverse operation keeps the minus sign: ∫e−x dx=−e−x+C\int e^{-x}\,dx = -e^{-x} + C.

Where the derivative of e^(-x) shows up on the AP exam

The chain rule is Topic 3.1, on both AB and BC, and e−xe^{-x} is the first composite most students meet. It returns in Unit 7, where y=e−xy = e^{-x} solves the decay equation dydx=−y\frac{dy}{dx} = -y.

Common mistakes with the derivative of e^(-x)

  • Answering e−xe^{-x}, forgetting the inner derivative −1-1 and losing the minus sign.
  • Answering −xe−x−1-xe^{-x-1}, applying the power rule to an exponential.
  • Applying the chain factor twice and writing −e−x⋅(−1)=e−x-e^{-x}\cdot(-1) = e^{-x}, which undoes the sign.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of e^(-x)?

ddxe−x=−e−x\frac{d}{dx}e^{-x} = -e^{-x}. The chain rule multiplies e−xe^{-x} by the derivative of the exponent −x-x, which is −1-1.

Why is there a minus sign?

The inner function −x-x has derivative −1-1, and the chain rule multiplies by it, flipping the sign of the positive quantity e−xe^{-x}.

What is the integral of e^(-x)?

∫e−x dx=−e−x+C\int e^{-x}\,dx = -e^{-x} + C. Differentiating −e−x-e^{-x} returns e−xe^{-x}, which confirms it.