Calculus Examples

$2^{\ln (x)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

$\frac{d}{d u} [2^{u}] \frac{d}{d x} [\ln (x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

$2^{u} \ln (2) \frac{d}{d x} [\ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $\ln (x)$ .

$2^{\ln (x)} \ln (2) \frac{d}{d x} [\ln (x)]$

Step 2

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

$2^{\ln (x)} \ln (2) \frac{1}{x}$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{1}{x}$ and $2^{\ln (x)}$ .

$\frac{2^{\ln (x)}}{x} \ln (2)$

Step 3.2

Combine $\frac{2^{\ln (x)}}{x}$ and $\ln (2)$ .

$\frac{2^{\ln (x)} \ln (2)}{x}$