Calculus Examples

$\log_{5} (x e^{x})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\log_{5} (u)] \frac{d}{d x} [x e^{x}]$

Step 1.2

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

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

Step 1.3

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

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

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$\frac{1}{x e^{x} \ln (5)} (x e^{x} + e^{x} \cdot 1)$

Step 4.2

Multiply $e^{x}$ by $1$ .

$\frac{1}{x e^{x} \ln (5)} (x e^{x} + e^{x})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{1}{x e^{x} \ln (5)} (x e^{x}) + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2

Combine terms.

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Step 5.2.1

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

$\frac{x}{x e^{x} \ln (5)} e^{x} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.2

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

$\frac{x e^{x}}{x e^{x} \ln (5)} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.3

Cancel the common factor of $x$ .

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Step 5.2.3.1

Cancel the common factor.

$\frac{x e^{x}}{x e^{x} \ln (5)} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.3.2

Rewrite the expression.

$\frac{e^{x}}{e^{x} \ln (5)} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.4

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

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Step 5.2.4.1

Cancel the common factor.

$\frac{e^{x}}{e^{x} \ln (5)} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.4.2

Rewrite the expression.

$\frac{1}{\ln (5)} + \frac{1}{x e^{x} \ln (5)} e^{x}$

Step 5.2.5

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

$\frac{1}{\ln (5)} + \frac{e^{x}}{x e^{x} \ln (5)}$

Step 5.2.6

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

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Step 5.2.6.1

Cancel the common factor.

$\frac{1}{\ln (5)} + \frac{e^{x}}{x e^{x} \ln (5)}$

Step 5.2.6.2

Rewrite the expression.

$\frac{1}{\ln (5)} + \frac{1}{x \ln (5)}$

Step 5.3

Reorder terms.

$\frac{1}{x \ln (5)} + \frac{1}{\ln (5)}$