Calculus Examples

$y = x^{5} \sin (x)$

Step 1

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

$\ln (y) = \ln (x^{5} \sin (x))$

Step 2

Expand the right hand side.

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Step 2.1

Rewrite $\ln (x^{5} \sin (x))$ as $\ln (x^{5}) + \ln (\sin (x))$ .

$\ln (y) = \ln (x^{5}) + \ln (\sin (x))$

Step 2.2

Expand $\ln (x^{5})$ by moving $5$ outside the logarithm.

$\ln (y) = 5 \ln (x) + \ln (\sin (x))$

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.

$\frac{y'}{y} = 5 \ln (x) + \ln (\sin (x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $5 \ln (x) + \ln (\sin (x))$ .

$\frac{y'}{y} = \frac{d}{d x} [5 \ln (x) + \ln (\sin (x))]$

Step 3.2.2

By the Sum Rule, the derivative of $5 \ln (x) + \ln (\sin (x))$ with respect to $x$ is $\frac{d}{d x} [5 \ln (x)] + \frac{d}{d x} [\ln (\sin (x))]$ .

$\frac{y'}{y} = \frac{d}{d x} [5 \ln (x)] + \frac{d}{d x} [\ln (\sin (x))]$

Step 3.2.3

Evaluate $\frac{d}{d x} [5 \ln (x)]$ .

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Step 3.2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \ln (x)$ with respect to $x$ is $5 \frac{d}{d x} [\ln (x)]$ .

$\frac{y'}{y} = 5 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (\sin (x))]$

Step 3.2.3.2

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

$\frac{y'}{y} = 5 \frac{1}{x} + \frac{d}{d x} [\ln (\sin (x))]$

Step 3.2.3.3

Combine $5$ and $\frac{1}{x}$ .

$\frac{y'}{y} = \frac{5}{x} + \frac{d}{d x} [\ln (\sin (x))]$

Step 3.2.4

Evaluate $\frac{d}{d x} [\ln (\sin (x))]$ .

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Step 3.2.4.1

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 3.2.4.1.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$\frac{y'}{y} = \frac{5}{x} + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sin (x)]$

Step 3.2.4.1.2

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

$\frac{y'}{y} = \frac{5}{x} + \frac{1}{u} \frac{d}{d x} [\sin (x)]$

Step 3.2.4.1.3

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

$\frac{y'}{y} = \frac{5}{x} + \frac{1}{\sin (x)} \frac{d}{d x} [\sin (x)]$

Step 3.2.4.2

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

$\frac{y'}{y} = \frac{5}{x} + \frac{1}{\sin (x)} \cos (x)$

Step 3.2.4.3

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

$\frac{y'}{y} = \frac{5}{x} + \csc (x) \cos (x)$

Step 3.2.5

Simplify.

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Step 3.2.5.1

Reorder terms.

$\frac{y'}{y} = \cos (x) \csc (x) + \frac{5}{x}$

Step 3.2.5.2

Simplify each term.

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Step 3.2.5.2.1

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

$\frac{y'}{y} = \cos (x) \frac{1}{\sin (x)} + \frac{5}{x}$

Step 3.2.5.2.2

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

$\frac{y'}{y} = \frac{\cos (x)}{\sin (x)} + \frac{5}{x}$

Step 3.2.5.3

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

$\frac{y'}{y} = \cot (x) + \frac{5}{x}$

Step 4

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

$y' = (\cot (x) + \frac{5}{x}) (x^{5} \sin (x))$

Step 5

Simplify the right hand side.

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Step 5.1

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

$y' = (\frac{\cos (x)}{\sin (x)} + \frac{5}{x}) (x^{5} \sin (x))$

Step 5.2

Apply the distributive property.

$y' = \frac{\cos (x)}{\sin (x)} (x^{5} \sin (x)) + \frac{5}{x} (x^{5} \sin (x))$

Step 5.3

Cancel the common factor of $\sin (x)$ .

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Step 5.3.1

Factor $\sin (x)$ out of $x^{5} \sin (x)$ .

$y' = \frac{\cos (x)}{\sin (x)} (\sin (x) x^{5}) + \frac{5}{x} (x^{5} \sin (x))$

Step 5.3.2

Cancel the common factor.

$y' = \frac{\cos (x)}{\sin (x)} (\sin (x) x^{5}) + \frac{5}{x} (x^{5} \sin (x))$

Step 5.3.3

Rewrite the expression.

$y' = \cos (x) x^{5} + \frac{5}{x} (x^{5} \sin (x))$

Step 5.4

Cancel the common factor of $x$ .

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Step 5.4.1

Factor $x$ out of $x^{5} \sin (x)$ .

$y' = \cos (x) x^{5} + \frac{5}{x} (x (x^{4} \sin (x)))$

Step 5.4.2

Cancel the common factor.

$y' = \cos (x) x^{5} + \frac{5}{x} (x (x^{4} \sin (x)))$

Step 5.4.3

Rewrite the expression.

$y' = \cos (x) x^{5} + 5 (x^{4} \sin (x))$

Step 5.5

Reorder factors in $\cos (x) x^{5} + 5 x^{4} \sin (x)$ .

$y' = x^{5} \cos (x) + 5 x^{4} \sin (x)$