Derivative of eˣ

\frac{d}{dx}\left[e^{x}\right] = e^{x}.

Differentiate again: \frac{d^2}{dx^2}\left[e^{x}\right] = e^{x}.

Plug into the derivative: f'(1) = 2.71828. That is the slope of the tangent line there; drag the graph to see it.

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

eˣ is the one function whose slope at every point equals its own height. That property is really the definition of the number e ≈ 2.71828.

Drag the graph · at x = 1.20 the slope is 3.3201, the height of the dashed curve.

Questions students ask

What is the derivative of eˣ?

$\frac{d}{dx}\left[e^{x}\right] = e^{x}$ .

What is the second derivative of eˣ?

Differentiate again: $\frac{d^2}{dx^2}\left[e^{x}\right] = e^{x}$ .

What is the slope of eˣ at x = 1?

Plug into the derivative: $f'(1) = 2.71828$ . That is the slope of the tangent line there; drag the graph to see it.

How do you find the derivative of eˣ?

eˣ is the one function whose slope at every point equals its own height. That property is really the definition of the number e ≈ 2.71828.

What is the integral of eˣ?

See the integral of eˣ page for the antiderivative with steps.