Calculus Examples

$y = {\sin (x)}^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({\sin (x)}^{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Differentiate using the Power Rule.

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Remove parentheses.

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

Step 3.8.3

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = e^{x \ln (\sin (x))} x \cos (x) \csc (x) + e^{x \ln (\sin (x))} \ln (\sin (x))$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = e^{x \ln (\sin (x))} x \cos (x) \csc (x) + e^{x \ln (\sin (x))} \ln (\sin (x))$