Calculus Examples

$\ln (\sin (x^{x}))$

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 $g (x) = \sin (x^{x})$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (x^{x})$ .

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

Step 2

Convert from $\frac{1}{\sin (x^{x})}$ to $\csc (x^{x})$ .

$\csc (x^{x}) \frac{d}{d x} [\sin (x^{x})]$

Step 3

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

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Step 3.1

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

$\csc (x^{x}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{x}])$

Step 3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\csc (x^{x}) (\cos (u_{2}) \frac{d}{d x} [x^{x}])$

Step 3.3

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

$\csc (x^{x}) (\cos (x^{x}) \frac{d}{d x} [x^{x}])$

Step 4

Use the properties of logarithms to simplify the differentiation.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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 5.1

To apply the Chain Rule, set $u_{3}$ as $x \ln (x)$ .

$\csc (x^{x}) \cos (x^{x}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [x \ln (x)])$

Step 5.2

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

$\csc (x^{x}) \cos (x^{x}) (e^{u_{3}} \frac{d}{d x} [x \ln (x)])$

Step 5.3

Replace all occurrences of $u_{3}$ with $x \ln (x)$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Differentiate using the Power Rule.

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Step 8.1

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

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

Step 8.2

Cancel the common factor of $x$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

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

$\csc (x^{x}) \cos (x^{x}) e^{x \ln (x)} + \csc (x^{x}) \cos (x^{x}) (e^{x \ln (x)} \ln (x))$

Step 9.4

Reorder terms.

$e^{x \ln (x)} \cos (x^{x}) \csc (x^{x}) + e^{x \ln (x)} \cos (x^{x}) \csc (x^{x}) \ln (x)$