Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

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

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

Step 1.2

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

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

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

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Step 2.1

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

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

Step 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 $a$ = $e$ .

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

Step 2.3

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

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

Step 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 $f (x) = \cos (x)$ and $g (x) = \ln (\sin (x))$ .

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

Step 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 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

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

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

Step 7

Raise $\cos (x)$ to the power of $1$ .

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

Step 8

Raise $\cos (x)$ to the power of $1$ .

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

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10

Add $1$ and $1$ .

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

Step 11

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

$e^{\cos (x) \ln (\sin (x))} (\cos^{2} (x) \csc (x) + \ln (\sin (x)) (- \sin (x)))$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$e^{\cos (x) \ln (\sin (x))} (\cos^{2} (x) \csc (x)) + e^{\cos (x) \ln (\sin (x))} (\ln (\sin (x)) (- \sin (x)))$

Step 12.2

Remove parentheses.

$e^{\cos (x) \ln (\sin (x))} \cos^{2} (x) \csc (x) + e^{\cos (x) \ln (\sin (x))} \ln (\sin (x)) (- \sin (x))$

Step 12.3

Reorder terms.

$e^{\cos (x) \ln (\sin (x))} \cos^{2} (x) \csc (x) - e^{\cos (x) \ln (\sin (x))} \sin (x) \ln (\sin (x))$