Calculus Examples

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

Step 1

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

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

Step 2

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

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

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.

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

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

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

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

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

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

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 $\sin (x)$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Combine $\ln (\sin (x))$ and $\frac{1}{x}$ .

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

Step 3.2.8

Simplify.

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Step 3.2.8.1

Reorder terms.

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

Step 3.2.8.2

Simplify each term.

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Step 3.2.8.2.1

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

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

Step 3.2.8.2.2

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 3.2.8.2.3

Combine $\frac{\cos (x)}{\sin (x)}$ and $\ln (x)$ .

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

Step 3.2.8.3

Simplify each term.

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Step 3.2.8.3.1

Separate fractions.

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

Step 3.2.8.3.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 3.2.8.3.3

Divide $\ln (x)$ by $1$ .

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

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = (\ln (x) \cot (x) + \frac{\ln (\sin (x))}{x}) {\sin (x)}^{\ln (x)}$

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.

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

Step 5.1.2

Combine $\ln (x)$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine $\frac{\ln (x) \cos (x)}{\sin (x)}$ and ${\sin (x)}^{\ln (x)}$ .

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

Step 5.4

Combine $\frac{\ln (\sin (x))}{x}$ and ${\sin (x)}^{\ln (x)}$ .

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

Step 5.5

Cancel the common factor of ${\sin (x)}^{\ln (x)}$ and $\sin (x)$ .

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Step 5.5.1

Factor $\sin (x)$ out of $\ln (x) \cos (x) {\sin (x)}^{\ln (x)}$ .

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

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Multiply by $1$ .

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

Step 5.5.2.2

Cancel the common factor.

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

Step 5.5.2.3

Rewrite the expression.

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

Step 5.5.2.4

Divide $\ln (x) \cos (x) {\sin (x)}^{\ln (x) - 1}$ by $1$ .

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

Step 5.6

To write $\ln (x) \cos (x) {\sin (x)}^{\ln (x) - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.7

Combine the numerators over the common denominator.

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

Step 5.8

Reorder factors in $\frac{\ln (x) \cos (x) {\sin (x)}^{\ln (x) - 1} x + \ln (\sin (x)) {\sin (x)}^{\ln (x)}}{x}$ .

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