Calculus Examples

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

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

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

Step 2

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

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

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ 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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