Derivative of ex

Derivative of e^x

How to solve the derivative of ex problems: formula, proof, examples, and their solutions.

Formula

[e^x]' = e^x

The derivative of ex is itself: ex.

Proof

Derivative of e^x: Proof of the Formula

ln both sides.

Natural logarithms

Differentiate both sides.

Implicit differentiation

Derivative of ln x

Multiply y on both sides.

Change y into ex.

So [ex]' = ex.

Example 1

Find the derivative of the given function. y = x^2 * e^x

y = (x2)(ex)

So y' is equal to,
the derivative of x2, 2x
times ex
plus x2,
times, the derivative of ex, ex.

Product rule in differentiation

Power rule in differentiation (Part 1)

Example 2

Find the derivative of the given function. y = e^(x^2 - 4x)

y = ex2 - 4x

So y' is equal to,
the derivative of the outer part, ex2 - 4x
times, the derivative of the inner part, (2x - 4).

Chain rule in differentiation

Derivative of polynomials