Derivative of -e^x
2026-02-28 10:28 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-minus-e-to-the-power-x

We can derive the derivative of -ex using proofs.

To show this, we will use the properties of exponential functions along with the basic rules of differentiation. There are several methods we use to prove this, such as:

By First Principle Using Constant Multiplier Rule

We will now demonstrate that the differentiation of -ex results in -e^x using the above-mentioned methods:

By First Principle The derivative of -ex can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of -e^x using the first principle, we will consider f(x) = -ex.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Given that f(x) = -ex, we write f(x + h) = -e(x + h).

Substituting these into the equation, f'(x) = limₕ→₀ [-e^(x + h) + e^x] / h = limₕ→₀ [-ex * (eh - 1)] / h

Using the limit properties, limₕ→₀ (eh - 1)/h = 1, f'(x) = -ex * 1 = -ex.

Hence, proved.

Using Constant Multiplier Rule

To prove the differentiation of -ex using the constant multiplier rule, We use the formula:

The derivative of a constant times a function is the constant times the derivative of the function.

Let u(x) = ex, so -ex = -1 * u(x).

By the constant multiplier rule: d/dx(-1 * u(x)) = -1 * d/dx(u(x)) Since d/dx(ex) = ex, we have: d/dx(-ex) = -1 * ex = -ex. Thus, the derivative is -ex.