Calculus Examples

$f (x) = 3 e^{x} \ln (x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x} \ln (x)$ with respect to $x$ is $3 \frac{d}{d x} [e^{x} \ln (x)]$ .

$3 \frac{d}{d x} [e^{x} \ln (x)]$

Step 2

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 $f (x) = e^{x}$ and $g (x) = \ln (x)$ .

$3 (e^{x} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [e^{x}])$

Step 3

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

$3 (e^{x} \frac{1}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 4

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

$3 (\frac{e^{x}}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 5

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

$3 (\frac{e^{x}}{x} + \ln (x) e^{x})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$3 \frac{e^{x}}{x} + 3 (\ln (x) e^{x})$

Step 6.2

Combine $3$ and $\frac{e^{x}}{x}$ .

$\frac{3 e^{x}}{x} + 3 \ln (x) e^{x}$

Step 6.3

Reorder terms.

$3 e^{x} \ln (x) + \frac{3 e^{x}}{x}$