Differentiating exponential functions

Differentiating e^f(x) with the chain rule, and functions involving e^x with the product and quotient rules.

The chain rule

Exponential of a function

$$\frac{d}{dx}e^{f(x)} = f'(x)\,e^{f(x)}$$

In words: the exponential stays the same, and it is multiplied by the derivative of the index.

Example 1●

Differentiate $y = e^{x^2 + 1}$.

$\dfrac{dy}{dx} = 2xe^{x^2 + 1}$

Example 2●●

Differentiate $y = (e^x + 1)^3$.

$\dfrac{dy}{dx} = 3(e^x + 1)^2 \times e^x = 3e^x(e^x + 1)^2$

Chain rule on the bracket

Example 3: product●●

Differentiate $y = x^2e^x$.

$u = x^2$, $v = e^x$: $\dfrac{dy}{dx} = 2xe^x + x^2e^x = x(x + 2)e^x$.

Example 4: quotient●●

Differentiate $y = \dfrac{e^x}{x}$.

$u = e^x$, $v = x$: $\dfrac{dy}{dx} = \dfrac{xe^x - e^x}{x^2} = \dfrac{(x - 1)e^x}{x^2}$.

Flashcards

$dfrac{d}{dx}e^{f(x)}$

$f'(x)\,e^{f(x)}$

$dfrac{d}{dx}xe^x$

$(x + 1)e^x$

$dfrac{d}{dx}dfrac{e^x}{x}$

$\dfrac{(x - 1)e^x}{x^2}$

Practice