Calculus Examples

$5 {\sin (x)}^{x}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 {\sin (x)}^{x}$ with respect to $x$ is $5 \frac{d}{d x} [{\sin (x)}^{x}]$ .

$5 \frac{d}{d x} [{\sin (x)}^{x}]$

Step 2

Use the properties of logarithms to simplify the differentiation.

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Step 2.1

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

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

Step 2.2

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

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

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

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Step 3.1

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

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

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 5.3

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

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

Step 6

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

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

Step 7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 8

Differentiate using the Power Rule.

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Step 8.1

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

$5 e^{x \ln (\sin (x))} (x (\csc (x) \cos (x)) + \ln (\sin (x)) \cdot 1)$

Step 8.2

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

$5 e^{x \ln (\sin (x))} (x (\csc (x) \cos (x)) + \ln (\sin (x)))$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$5 e^{x \ln (\sin (x))} (x (\csc (x) \cos (x))) + 5 e^{x \ln (\sin (x))} \ln (\sin (x))$

Step 9.2

Remove parentheses.

$5 e^{x \ln (\sin (x))} x \csc (x) \cos (x) + 5 e^{x \ln (\sin (x))} \ln (\sin (x))$

Step 9.3

Reorder terms.

$5 e^{x \ln (\sin (x))} x \cos (x) \csc (x) + 5 e^{x \ln (\sin (x))} \ln (\sin (x))$