Calculus Examples

$\ln (x^{x})$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [x^{x}]$

Step 1.3

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

$\frac{1}{x^{x}} \frac{d}{d x} [x^{x}]$

Step 2

Use the properties of logarithms to simplify the differentiation.

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Step 2.1

Rewrite $x^{x}$ as $e^{\ln (x^{x})}$ .

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

Step 2.2

Expand $\ln (x^{x})$ by moving $x$ outside the logarithm.

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

Step 3

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 $g (x) = x \ln (x)$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x \ln (x)$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $x \ln (x)$ .

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

Step 4

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

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

Step 5

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 $g (x) = \ln (x)$ .

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

Step 6

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

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

Step 7

Differentiate using the Power Rule.

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Step 7.1

Combine $x$ and $\frac{1}{x}$ .

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

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 7.4

Multiply $\ln (x)$ by $1$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Combine terms.

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Step 8.2.1

Multiply $\frac{e^{x \ln (x)}}{x^{x}}$ by $1$ .

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

Step 8.2.2

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

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