Calculus Examples

$y = {\sin (7 x)}^{\ln (x)}$

Step 1

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

$\ln (y) = \ln ({\sin (7 x)}^{\ln (x)})$

Step 2

Expand $\ln ({\sin (7 x)}^{\ln (x)})$ by moving $\ln (x)$ outside the logarithm.

$\ln (y) = \ln (x) \ln (\sin (7 x))$

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.

$\frac{y'}{y} = \ln (x) \ln (\sin (7 x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $\ln (x) \ln (\sin (7 x))$ .

$\frac{y'}{y} = \frac{d}{d x} [\ln (x) \ln (\sin (7 x))]$

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 $g (x) = \ln (\sin (7 x))$ .

$\frac{y'}{y} = \ln (x) \frac{d}{d x} [\ln (\sin (7 x))] + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

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 $g (x) = \sin (7 x)$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $\sin (7 x)$ .

$\frac{y'}{y} = \ln (x) (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\sin (7 x)]) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.3.2

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

$\frac{y'}{y} = \ln (x) (\frac{1}{u_{1}} \frac{d}{d x} [\sin (7 x)]) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $\sin (7 x)$ .

$\frac{y'}{y} = \ln (x) (\frac{1}{\sin (7 x)} \frac{d}{d x} [\sin (7 x)]) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.4

Convert from $\frac{1}{\sin (7 x)}$ to $\csc (7 x)$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) \frac{d}{d x} [\sin (7 x)]) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 7 x$ .

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Step 3.2.5.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [7 x])) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.5.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) (\cos (u_{2}) \frac{d}{d x} [7 x])) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.5.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) (\cos (7 x) \frac{d}{d x} [7 x])) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.6

Differentiate.

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Step 3.2.6.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) \cos (7 x) (7 \frac{d}{d x} [x])) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.6.2

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

$\frac{y'}{y} = \ln (x) (\csc (7 x) \cos (7 x) (7 \cdot 1)) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.6.3

Simplify the expression.

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Step 3.2.6.3.1

Multiply $7$ by $1$ .

$\frac{y'}{y} = \ln (x) (\csc (7 x) \cos (7 x) \cdot 7) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.6.3.2

Move $7$ to the left of $\csc (7 x) \cos (7 x)$ .

$\frac{y'}{y} = \ln (x) (7 \cdot (\csc (7 x) \cos (7 x))) + \ln (\sin (7 x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.7

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

$\frac{y'}{y} = \ln (x) (7 \csc (7 x) \cos (7 x)) + \ln (\sin (7 x)) \frac{1}{x}$

Step 3.2.8

Combine $\ln (\sin (7 x))$ and $\frac{1}{x}$ .

$\frac{y'}{y} = \ln (x) (7 \csc (7 x) \cos (7 x)) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9

Simplify.

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Step 3.2.9.1

Reorder terms.

$\frac{y'}{y} = 7 \cos (7 x) \csc (7 x) \ln (x) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.2

Simplify each term.

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Step 3.2.9.2.1

Rewrite $\csc (7 x)$ in terms of sines and cosines.

$\frac{y'}{y} = 7 \cos (7 x) \frac{1}{\sin (7 x)} \ln (x) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.2.2

Multiply $7 \cos (7 x) \frac{1}{\sin (7 x)}$ .

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Step 3.2.9.2.2.1

Combine $\frac{1}{\sin (7 x)}$ and $7$ .

$\frac{y'}{y} = \frac{7}{\sin (7 x)} \cos (7 x) \ln (x) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.2.2.2

Combine $\frac{7}{\sin (7 x)}$ and $\cos (7 x)$ .

$\frac{y'}{y} = \frac{7 \cos (7 x)}{\sin (7 x)} \ln (x) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.2.3

Combine $\frac{7 \cos (7 x)}{\sin (7 x)}$ and $\ln (x)$ .

$\frac{y'}{y} = \frac{7 \cos (7 x) \ln (x)}{\sin (7 x)} + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.3

Simplify each term.

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Step 3.2.9.3.1

Separate fractions.

$\frac{y'}{y} = \frac{7 \ln (x)}{1} \cdot \frac{\cos (7 x)}{\sin (7 x)} + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.3.2

Convert from $\frac{\cos (7 x)}{\sin (7 x)}$ to $\cot (7 x)$ .

$\frac{y'}{y} = \frac{7 \ln (x)}{1} \cot (7 x) + \frac{\ln (\sin (7 x))}{x}$

Step 3.2.9.3.3

Divide $7 \ln (x)$ by $1$ .

$\frac{y'}{y} = 7 \ln (x) \cot (7 x) + \frac{\ln (\sin (7 x))}{x}$

Step 4

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

$y' = (7 \ln (x) \cot (7 x) + \frac{\ln (\sin (7 x))}{x}) {\sin (7 x)}^{\ln (x)}$

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

Simplify $7 \ln (x)$ by moving $7$ inside the logarithm.

$y' = (\ln (x^{7}) \cot (7 x) + \frac{\ln (\sin (7 x))}{x}) {\sin (7 x)}^{\ln (x)}$

Step 5.1.2

Rewrite $\cot (7 x)$ in terms of sines and cosines.

$y' = (\ln (x^{7}) \frac{\cos (7 x)}{\sin (7 x)} + \frac{\ln (\sin (7 x))}{x}) {\sin (7 x)}^{\ln (x)}$

Step 5.1.3

Combine $\ln (x^{7})$ and $\frac{\cos (7 x)}{\sin (7 x)}$ .

$y' = (\frac{\ln (x^{7}) \cos (7 x)}{\sin (7 x)} + \frac{\ln (\sin (7 x))}{x}) {\sin (7 x)}^{\ln (x)}$

Step 5.2

Apply the distributive property.

$y' = \frac{\ln (x^{7}) \cos (7 x)}{\sin (7 x)} {\sin (7 x)}^{\ln (x)} + \frac{\ln (\sin (7 x))}{x} {\sin (7 x)}^{\ln (x)}$

Step 5.3

Combine $\frac{\ln (x^{7}) \cos (7 x)}{\sin (7 x)}$ and ${\sin (7 x)}^{\ln (x)}$ .

$y' = \frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x)}}{\sin (7 x)} + \frac{\ln (\sin (7 x))}{x} {\sin (7 x)}^{\ln (x)}$

Step 5.4

Combine $\frac{\ln (\sin (7 x))}{x}$ and ${\sin (7 x)}^{\ln (x)}$ .

$y' = \frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x)}}{\sin (7 x)} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.5

Cancel the common factor of ${\sin (7 x)}^{\ln (x)}$ and $\sin (7 x)$ .

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Step 5.5.1

Factor $\sin (7 x)$ out of $\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x)}$ .

$y' = \frac{\sin (7 x) (\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1})}{\sin (7 x)} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Multiply by $1$ .

$y' = \frac{\sin (7 x) (\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1})}{\sin (7 x) \cdot 1} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.5.2.2

Cancel the common factor.

$y' = \frac{\sin (7 x) (\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1})}{\sin (7 x) \cdot 1} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.5.2.3

Rewrite the expression.

$y' = \frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1}}{1} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.5.2.4

Divide $\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1}$ by $1$ .

$y' = \ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.6

To write $\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = \frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1} x}{x} + \frac{\ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.7

Combine the numerators over the common denominator.

$y' = \frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1} x + \ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$

Step 5.8

Reorder factors in $\frac{\ln (x^{7}) \cos (7 x) {\sin (7 x)}^{\ln (x) - 1} x + \ln (\sin (7 x)) {\sin (7 x)}^{\ln (x)}}{x}$ .

$y' = \frac{x {\sin (7 x)}^{\ln (x) - 1} \ln (x^{7}) \cos (7 x) + {\sin (7 x)}^{\ln (x)} \ln (\sin (7 x))}{x}$