Calculus Examples

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

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

Step 1

Let

y = f (x)

$y = f (x)$ , take the natural logarithm of both sides

ln (y) = ln (f (x))

$\ln (y) = \ln (f (x))$ .

ln (y) = ln ({(sin (x))}^{cos (x)})

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

Step 2

Expand

ln ({(sin (x))}^{cos (x)})

$\ln ({(\sin (x))}^{\cos (x)})$ by moving

cos (x)

$\cos (x)$ outside the logarithm.

ln (y) = cos (x) ln (sin (x))

$\ln (y) = \cos (x) \ln (\sin (x))$

Step 3

Differentiate the expression using the chain rule, keeping in mind that

y

$y$ is a function of

x

$x$ .

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Step 3.1

Differentiate the left hand side

ln (y)

$\ln (y)$ using the chain rule.

$\frac{y'}{y} = \cos (x) \ln (\sin (x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate

$\cos (x) \ln (\sin (x))$ .

$\frac{y'}{y} = \frac{d}{d x} [\cos (x) \ln (\sin (x))]$

Step 3.2.2

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

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

$\frac{y'}{y} = \cos (x) \frac{d}{d x} [\ln (\sin (x))] + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \ln (x)$ and

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

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Step 3.2.3.1

To apply the Chain Rule, set

$u$ as

$\sin (x)$ .

$\frac{y'}{y} = \cos (x) (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sin (x)]) + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.3.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

$\frac{y'}{y} = \cos (x) (\frac{1}{u} \frac{d}{d x} [\sin (x)]) + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.3.3

Replace all occurrences of

$u$ with

$\sin (x)$ .

$\frac{y'}{y} = \cos (x) (\frac{1}{\sin (x)} \frac{d}{d x} [\sin (x)]) + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.4

Convert from

$\frac{1}{\sin (x)}$ to

$\csc (x)$ .

$\frac{y'}{y} = \cos (x) (\csc (x) \frac{d}{d x} [\sin (x)]) + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.5

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

$\frac{y'}{y} = \cos (x) \csc (x) \cos (x) + \ln (\sin (x)) \frac{d}{d x} [\cos (x)]$

Step 3.2.6

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 3.2.7

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 3.2.8

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.9

Add

$1$ and

$1$ .

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

Step 3.2.10

The derivative of

$\cos (x)$ with respect to

$x$ is

$- \sin (x)$ .

$\frac{y'}{y} = \cos^{2} (x) \csc (x) + \ln (\sin (x)) (- \sin (x))$

Step 3.2.11

Simplify.

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Step 3.2.11.1

Reorder terms.

$\frac{y'}{y} = \cos^{2} (x) \csc (x) - \sin (x) \ln (\sin (x))$

Step 3.2.11.2

Simplify each term.

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Step 3.2.11.2.1

Rewrite

$\csc (x)$ in terms of sines and cosines.

$\frac{y'}{y} = \cos^{2} (x) \frac{1}{\sin (x)} - \sin (x) \ln (\sin (x))$

Step 3.2.11.2.2

Combine

$\cos^{2} (x)$ and

$\frac{1}{\sin (x)}$ .

$\frac{y'}{y} = \frac{\cos^{2} (x)}{\sin (x)} - \sin (x) \ln (\sin (x))$

Step 3.2.11.3

Simplify each term.

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Step 3.2.11.3.1

Factor

$\cos (x)$ out of

$\cos^{2} (x)$ .

$\frac{y'}{y} = \frac{\cos (x) \cos (x)}{\sin (x)} - \sin (x) \ln (\sin (x))$

Step 3.2.11.3.2

Separate fractions.

$\frac{y'}{y} = \frac{\cos (x)}{1} \cdot \frac{\cos (x)}{\sin (x)} - \sin (x) \ln (\sin (x))$

Step 3.2.11.3.3

Convert from

$\frac{\cos (x)}{\sin (x)}$ to

$\cot (x)$ .

$\frac{y'}{y} = \frac{\cos (x)}{1} \cot (x) - \sin (x) \ln (\sin (x))$

Step 3.2.11.3.4

Divide

$\cos (x)$ by

$1$ .

$\frac{y'}{y} = \cos (x) \cot (x) - \sin (x) \ln (\sin (x))$

Step 4

Isolate

$y'$ and substitute the original function for

$y$ in the right hand side.

$y' = (\cos (x) \cot (x) - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5

Simplify the right hand side.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite

$\cot (x)$ in terms of sines and cosines.

$y' = (\cos (x) \frac{\cos (x)}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.1.2

Multiply

$\cos (x) \frac{\cos (x)}{\sin (x)}$ .

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Step 5.1.2.1

Combine

$\cos (x)$ and

$\frac{\cos (x)}{\sin (x)}$ .

$y' = (\frac{\cos (x) \cos (x)}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.1.2.2

Raise

$\cos (x)$ to the power of

$1$ .

$y' = (\frac{\cos^{1} (x) \cos (x)}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.1.2.3

Raise

$\cos (x)$ to the power of

$1$ .

$y' = (\frac{\cos^{1} (x) \cos^{1} (x)}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.1.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = (\frac{{\cos (x)}^{1 + 1}}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.1.2.5

Add

$1$ and

$1$ .

$y' = (\frac{\cos^{2} (x)}{\sin (x)} - \sin (x) \ln (\sin (x))) {(\sin (x))}^{\cos (x)}$

Step 5.2

Apply the distributive property.

$y' = \frac{\cos^{2} (x)}{\sin (x)} {\sin (x)}^{\cos (x)} - \sin (x) \ln (\sin (x)) {\sin (x)}^{\cos (x)}$

Step 5.3

Combine

$\frac{\cos^{2} (x)}{\sin (x)}$ and

${\sin (x)}^{\cos (x)}$ .

$y' = \frac{\cos^{2} (x) {\sin (x)}^{\cos (x)}}{\sin (x)} - \sin (x) \ln (\sin (x)) {\sin (x)}^{\cos (x)}$

Step 5.4

Multiply

$\sin (x)$ by

${\sin (x)}^{\cos (x)}$ by adding the exponents.

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Step 5.4.1

Move

${\sin (x)}^{\cos (x)}$ .

$y' = \frac{\cos^{2} (x) {\sin (x)}^{\cos (x)}}{\sin (x)} - ({\sin (x)}^{\cos (x)} \sin (x)) \ln (\sin (x))$

Step 5.4.2

Multiply

${\sin (x)}^{\cos (x)}$ by

$\sin (x)$ .

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Step 5.4.2.1

Raise

$\sin (x)$ to the power of

$1$ .

$y' = \frac{\cos^{2} (x) {\sin (x)}^{\cos (x)}}{\sin (x)} - ({\sin (x)}^{\cos (x)} \sin^{1} (x)) \ln (\sin (x))$

Step 5.4.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = \frac{\cos^{2} (x) {\sin (x)}^{\cos (x)}}{\sin (x)} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.5

Cancel the common factor of

${\sin (x)}^{\cos (x)}$ and

$\sin (x)$ .

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Step 5.5.1

Factor

$\sin (x)$ out of

$\cos^{2} (x) {\sin (x)}^{\cos (x)}$ .

$y' = \frac{\sin (x) (\cos^{2} (x) {\sin (x)}^{\cos (x) - 1})}{\sin (x)} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Multiply by

$1$ .

$y' = \frac{\sin (x) (\cos^{2} (x) {\sin (x)}^{\cos (x) - 1})}{\sin (x) \cdot 1} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.5.2.2

Cancel the common factor.

$y' = \frac{\sin (x) (\cos^{2} (x) {\sin (x)}^{\cos (x) - 1})}{\sin (x) \cdot 1} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.5.2.3

Rewrite the expression.

$y' = \frac{\cos^{2} (x) {\sin (x)}^{\cos (x) - 1}}{1} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.5.2.4

Divide

$\cos^{2} (x) {\sin (x)}^{\cos (x) - 1}$ by

$1$ .

$y' = \cos^{2} (x) {\sin (x)}^{\cos (x) - 1} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

Step 5.6

Reorder factors in

$\cos^{2} (x) {\sin (x)}^{\cos (x) - 1} - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$ .

$y' = {\sin (x)}^{\cos (x) - 1} \cos^{2} (x) - {\sin (x)}^{\cos (x) + 1} \ln (\sin (x))$

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