Calculus Examples

$f (x) = \ln (\sin^{2} (x))$

Step 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 $g (x) = \sin^{2} (x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin^{2} (x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin^{2} (x)$ .

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

Step 2

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

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

Step 3.3

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

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

Step 4

Move $2$ to the left of $\csc^{2} (x)$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Reorder the factors of $2 \csc^{2} (x) \sin (x) \cos (x)$ .

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

Step 6.2

Add parentheses.

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

Step 6.3

Reorder $2 \csc^{2} (x)$ and $\cos (x) \sin (x)$ .

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

Step 6.4

Add parentheses.

$\cos (x) (\sin (x) \cdot 2) \csc^{2} (x)$

Step 6.5

Reorder $\cos (x)$ and $\sin (x) \cdot 2$ .

$\sin (x) \cdot 2 \cos (x) \csc^{2} (x)$

Step 6.6

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x) \csc^{2} (x)$

Step 6.7

Apply the sine double-angle identity.

$\sin (2 x) \csc^{2} (x)$

Step 6.8

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

$\sin (2 x) {(\frac{1}{\sin (x)})}^{2}$

Step 6.9

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\sin (2 x) \frac{1^{2}}{\sin^{2} (x)}$

Step 6.10

One to any power is one.

$\sin (2 x) \frac{1}{\sin^{2} (x)}$

Step 6.11

Combine $\sin (2 x)$ and $\frac{1}{\sin^{2} (x)}$ .

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

Step 6.12

Apply the sine double-angle identity.

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

Step 6.13

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

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Step 6.13.1

Factor $\sin (x)$ out of $2 \sin (x) \cos (x)$ .

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

Step 6.13.2

Cancel the common factors.

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Step 6.13.2.1

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

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

Step 6.13.2.2

Cancel the common factor.

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

Step 6.13.2.3

Rewrite the expression.

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

Step 6.14

Separate fractions.

$\frac{2}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Step 6.15

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

$\frac{2}{1} \cot (x)$

Step 6.16

Divide $2$ by $1$ .

$2 \cot (x)$