Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.1.1

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

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| \sin (x) |]$

Step 3.1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [| \sin (x) |]$

Step 3.1.3

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

$\frac{1}{| \sin (x) |} \frac{d}{d x} [| \sin (x) |]$

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (x)$ .

$\frac{1}{| \sin (x) |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [\sin (x)])$

Step 3.2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{| \sin (x) |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [\sin (x)])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

$\frac{1}{| \sin (x) |} (\frac{\sin (x)}{| \sin (x) |} \frac{d}{d x} [\sin (x)])$

Step 3.3

Multiply $\frac{\sin (x)}{| \sin (x) |}$ by $\frac{1}{| \sin (x) |}$ .

$\frac{\sin (x)}{| \sin (x) | | \sin (x) |} \frac{d}{d x} [\sin (x)]$

Step 3.4

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{\sin (x)}{| \sin (x) \sin (x) |} \frac{d}{d x} [\sin (x)]$

Step 3.5

Raise $\sin (x)$ to the power of $1$ .

$\frac{\sin (x)}{| \sin^{1} (x) \sin (x) |} \frac{d}{d x} [\sin (x)]$

Step 3.6

Raise $\sin (x)$ to the power of $1$ .

$\frac{\sin (x)}{| \sin^{1} (x) \sin^{1} (x) |} \frac{d}{d x} [\sin (x)]$

Step 3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x)}{| {\sin (x)}^{1 + 1} |} \frac{d}{d x} [\sin (x)]$

Step 3.8

Add $1$ and $1$ .

$\frac{\sin (x)}{| \sin^{2} (x) |} \frac{d}{d x} [\sin (x)]$

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Simplify.

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Step 3.11.1

Reorder terms.

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

Step 3.11.2

Remove non-negative terms from the absolute value.

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

Step 3.11.3

Cancel the common factor of $\sin (x)$ and $\sin^{2} (x)$ .

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Step 3.11.3.1

Factor $\sin (x)$ out of $\cos (x) \sin (x)$ .

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

Step 3.11.3.2

Cancel the common factors.

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Step 3.11.3.2.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 3.11.3.2.2

Cancel the common factor.

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

Step 3.11.3.2.3

Rewrite the expression.

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

Step 3.11.4

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

$\cot (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \cot (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \cot (x)$