Calculus Examples

$y = \log_{5} (e^{2 x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{5} (x)$ and $g (x) = e^{2 x}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $e^{2 x}$ .

$\frac{d}{d u_{1}} [\log_{5} (u_{1})] \frac{d}{d x} [e^{2 x}]$

Step 1.2

The derivative of $\log_{5} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (5)}$ .

$\frac{1}{u_{1} \ln (5)} \frac{d}{d x} [e^{2 x}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $e^{2 x}$ .

$\frac{1}{e^{2 x} \ln (5)} \frac{d}{d x} [e^{2 x}]$

Step 2

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 $g (x) = 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$\frac{1}{e^{2 x} \ln (5)} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [2 x])$

Step 2.2

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

$\frac{1}{e^{2 x} \ln (5)} (e^{u_{2}} \frac{d}{d x} [2 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$\frac{1}{e^{2 x} \ln (5)} (e^{2 x} \frac{d}{d x} [2 x])$

Step 3

Differentiate.

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Step 3.1

Combine $e^{2 x}$ and $\frac{1}{e^{2 x} \ln (5)}$ .

$\frac{e^{2 x}}{e^{2 x} \ln (5)} \frac{d}{d x} [2 x]$

Step 3.2

Cancel the common factor of $e^{2 x}$ .

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Step 3.2.1

Cancel the common factor.

$\frac{e^{2 x}}{e^{2 x} \ln (5)} \frac{d}{d x} [2 x]$

Step 3.2.2

Rewrite the expression.

$\frac{1}{\ln (5)} \frac{d}{d x} [2 x]$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{1}{\ln (5)} (2 \frac{d}{d x} [x])$

Step 3.4

Combine $2$ and $\frac{1}{\ln (5)}$ .

$\frac{2}{\ln (5)} \frac{d}{d x} [x]$

Step 3.5

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

$\frac{2}{\ln (5)} \cdot 1$

Step 3.6

Multiply $\frac{2}{\ln (5)}$ by $1$ .

$\frac{2}{\ln (5)}$