Calculus Examples

$y = 3^{\ln (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3^{\ln (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 3^{x}$ and $g (x) = \ln (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $3$ .

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

Step 3.1.3

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

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

Step 3.2

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

$3^{\ln (x)} \ln (3) \frac{1}{x}$

Step 3.3

Combine fractions.

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Step 3.3.1

Combine $\frac{1}{x}$ and $3^{\ln (x)}$ .

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

Step 3.3.2

Combine $\frac{3^{\ln (x)}}{x}$ and $\ln (3)$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3^{\ln (x)} \ln (3)}{x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3^{\ln (x)} \ln (3)}{x}$