Derivative of e^(-x)

Answer

-e^{-x}

Derivative of e^(-x)

Rule Used: Chain Rule
Steps: Identify a composite function in $e^{-x}$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: - $e^{-x}$

d/dx e^(-x) = -e^(-x). The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

Instant step-by-step solution for e^(-x)

Derivative ExponentialIntermediate

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Derivative of e^(-x) = -e^(-x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-e-to-the-minus-x

Derivative

e^(-x) → -e^(-x)engine: -e^(-x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

e^(−x) is a decaying exponential. The outer derivative of e^u is e^u, and u = −x has derivative −1, so the chain rule multiplies by −1. The function and its derivative are opposites: the slope is negative wherever the function is positive, which is everywhere. Forgetting the minus describes growth instead of decay.

Apply the chain rule to e^(-x). The form to remember, already simplified, is -e^(-x).

The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

At x = 0, substitute into the simplified form -e^(-x). That number is the slope of e^(-x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = e^(-x).
Apply the chain rule to e^(-x). The form to remember, already simplified, is -e^(-x).
At x = 0, substitute into the simplified form -e^(-x). That number is the slope of e^(-x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering e^(-x) instead of -e^(-x)

e^(−x) without the minus copies the function. Decay means the slope has the opposite sign of the height.

Treating e^(-x) as if only the base mattered

Only e^x is its own derivative with no extra factor. A coefficient in the exponent, or a base other than e, changes the answer.

Chain rule — The rule page lists this example and the others in the same family.
Exponential derivatives — The rule page lists this example and the others in the same family.
Derivative of e^x — Same family of differentiation, different function.
Derivative of e^(2x) — Same family of differentiation, different function.
Derivative of 2^x — Same family of differentiation, different function.
Derivative of 10^x — Same family of differentiation, different function.

Frequently asked questions

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

-e^(-x). e^(−x) is a decaying exponential. The outer derivative of e^u is e^u, and u = −x has derivative −1, so the chain rule multiplies by −1. The function and its derivative are opposites: the slope is negative wherever the function is positive, which is everywhere. Forgetting the minus describes growth instead of decay.