Algebra Examples

$\ln (x^{3} \sin^{2} (x))$

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 $g (x) = x^{3} \sin^{2} (x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $x^{3} \sin^{2} (x)$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{3} \sin^{2} (x)]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [x^{3} \sin^{2} (x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $x^{3} \sin^{2} (x)$ .

$\frac{1}{x^{3} \sin^{2} (x)} \frac{d}{d x} [x^{3} \sin^{2} (x)]$

Step 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 $f (x) = x^{3}$ and $g (x) = \sin^{2} (x)$ .

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} \frac{d}{d x} [\sin^{2} (x)] + \sin^{2} (x) \frac{d}{d x} [x^{3}])$

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 $\sin (x)$ .

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sin (x)]) + \sin^{2} (x) \frac{d}{d x} [x^{3}])$

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 $n = 2$ .

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (2 u_{2} \frac{d}{d x} [\sin (x)]) + \sin^{2} (x) \frac{d}{d x} [x^{3}])$

Step 3.3

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

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (2 \sin (x) \frac{d}{d x} [\sin (x)]) + \sin^{2} (x) \frac{d}{d x} [x^{3}])$

Step 4

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

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (2 \sin (x) \cos (x)) + \sin^{2} (x) \frac{d}{d x} [x^{3}])$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (2 \sin (x) \cos (x)) + \sin^{2} (x) (3 x^{2}))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{x^{3} \sin^{2} (x)} (x^{3} (2 \sin (x) \cos (x))) + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2

Combine terms.

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Step 6.2.1

Combine $x^{3}$ and $\frac{1}{x^{3} \sin^{2} (x)}$ .

$\frac{x^{3}}{x^{3} \sin^{2} (x)} (2 \sin (x) \cos (x)) + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.2

Combine $2$ and $\frac{x^{3}}{x^{3} \sin^{2} (x)}$ .

$\frac{2 x^{3}}{x^{3} \sin^{2} (x)} (\sin (x) \cos (x)) + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.3

Combine $\sin (x)$ and $\frac{2 x^{3}}{x^{3} \sin^{2} (x)}$ .

$\frac{\sin (x) (2 x^{3})}{x^{3} \sin^{2} (x)} \cos (x) + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.4

Combine $\frac{\sin (x) (2 x^{3})}{x^{3} \sin^{2} (x)}$ and $\cos (x)$ .

$\frac{\sin (x) (2 x^{3}) \cos (x)}{x^{3} \sin^{2} (x)} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.5

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

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Step 6.2.5.1

Factor $\sin (x)$ out of $\sin (x) (2 x^{3}) \cos (x)$ .

$\frac{\sin (x) ((2 x^{3}) \cos (x))}{x^{3} \sin^{2} (x)} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.5.2

Cancel the common factors.

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Step 6.2.5.2.1

Factor $\sin (x)$ out of $x^{3} \sin^{2} (x)$ .

$\frac{\sin (x) ((2 x^{3}) \cos (x))}{\sin (x) (x^{3} \sin (x))} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.5.2.2

Cancel the common factor.

$\frac{\sin (x) ((2 x^{3}) \cos (x))}{\sin (x) (x^{3} \sin (x))} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.5.2.3

Rewrite the expression.

$\frac{(2 x^{3}) \cos (x)}{x^{3} \sin (x)} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.6

Cancel the common factor of $x^{3}$ .

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Step 6.2.6.1

Cancel the common factor.

$\frac{2 x^{3} \cos (x)}{x^{3} \sin (x)} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.6.2

Rewrite the expression.

$\frac{(2) \cos (x)}{\sin (x)} + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.7

Convert from $\frac{2 \cos (x)}{\sin (x)}$ to $2 \cot (x)$ .

$2 \cot (x) + \frac{1}{x^{3} \sin^{2} (x)} (\sin^{2} (x) (3 x^{2}))$

Step 6.2.8

Combine $\sin^{2} (x)$ and $\frac{1}{x^{3} \sin^{2} (x)}$ .

$2 \cot (x) + \frac{\sin^{2} (x)}{x^{3} \sin^{2} (x)} (3 x^{2})$

Step 6.2.9

Combine $3$ and $\frac{\sin^{2} (x)}{x^{3} \sin^{2} (x)}$ .

$2 \cot (x) + \frac{3 \sin^{2} (x)}{x^{3} \sin^{2} (x)} x^{2}$

Step 6.2.10

Combine $\frac{3 \sin^{2} (x)}{x^{3} \sin^{2} (x)}$ and $x^{2}$ .

$2 \cot (x) + \frac{3 \sin^{2} (x) x^{2}}{x^{3} \sin^{2} (x)}$

Step 6.2.11

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

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Step 6.2.11.1

Cancel the common factor.

$2 \cot (x) + \frac{3 \sin^{2} (x) x^{2}}{x^{3} \sin^{2} (x)}$

Step 6.2.11.2

Rewrite the expression.

$2 \cot (x) + \frac{3 x^{2}}{x^{3}}$

Step 6.2.12

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 6.2.12.1

Factor $x^{2}$ out of $3 x^{2}$ .

$2 \cot (x) + \frac{x^{2} \cdot 3}{x^{3}}$

Step 6.2.12.2

Cancel the common factors.

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Step 6.2.12.2.1

Factor $x^{2}$ out of $x^{3}$ .

$2 \cot (x) + \frac{x^{2} \cdot 3}{x^{2} x}$

Step 6.2.12.2.2

Cancel the common factor.

$2 \cot (x) + \frac{x^{2} \cdot 3}{x^{2} x}$

Step 6.2.12.2.3

Rewrite the expression.

$2 \cot (x) + \frac{3}{x}$

Step 6.3

Reorder terms.

$\frac{3}{x} + 2 \cot (x)$