Find the Derivative - d/dx e^(8x)

$e^{8 x}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 8 x$ .

To apply the Chain Rule, set $u$ as $8 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [8 x]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [8 x]$

Replace all occurrences of $u$ with $8 x$ .

$e^{8 x} \frac{d}{d x} [8 x]$

Step 2

Differentiate.

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$e^{8 x} (8 \frac{d}{d x} [x])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{8 x} (8 \cdot 1)$

Simplify the expression.

Multiply $8$ by $1$ .

$e^{8 x} \cdot 8$

Move $8$ to the left of $e^{8 x}$ .

$8 e^{8 x}$