Calculus Examples

$\log_{3} (e^{x})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $e^{x}$ .

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

Step 1.2

The derivative of $\log_{3} (u)$ with respect to $u$ is $\frac{1}{u \ln (3)}$ .

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

Step 1.3

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $e^{x}$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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