Calculus Examples

$e^{5 x} \ln (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{5 x}$ and $g (x) = \ln (x)$ .

$e^{5 x} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [e^{5 x}]$

Step 2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$e^{5 x} \frac{1}{x} + \ln (x) \frac{d}{d x} [e^{5 x}]$

Step 3

Combine $e^{5 x}$ and $\frac{1}{x}$ .

$\frac{e^{5 x}}{x} + \ln (x) \frac{d}{d x} [e^{5 x}]$

Step 4

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 $g (x) = 5 x$ .

Tap for more steps...

To apply the Chain Rule, set $u$ as $5 x$ .

$\frac{e^{5 x}}{x} + \ln (x) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x])$

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

$\frac{e^{5 x}}{x} + \ln (x) (e^{u} \frac{d}{d x} [5 x])$

Replace all occurrences of $u$ with $5 x$ .

$\frac{e^{5 x}}{x} + \ln (x) (e^{5 x} \frac{d}{d x} [5 x])$

Step 5

Differentiate.

Tap for more steps...

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{e^{5 x}}{x} + \ln (x) (e^{5 x} (5 \frac{d}{d x} [x]))$

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

$\frac{e^{5 x}}{x} + \ln (x) (e^{5 x} (5 \cdot 1))$

Simplify the expression.

Tap for more steps...

Multiply $5$ by $1$ .

$\frac{e^{5 x}}{x} + \ln (x) (e^{5 x} \cdot 5)$

Move $5$ to the left of $e^{5 x}$ .

$\frac{e^{5 x}}{x} + \ln (x) (5 \cdot e^{5 x})$

Reorder terms.

$5 e^{5 x} \ln (x) + \frac{e^{5 x}}{x}$