Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = x^{3} - 2 x$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{3} - 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x]$ .

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

Step 3.2

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

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

Step 3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $- 2$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

$\frac{x^{3} \cos (x) - 2 x \cos (x) - \sin (x) (3 x^{2}) - \sin (x) \cdot - 2}{{(x^{3} - 2 x)}^{2}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $3$ by $- 1$ .

$\frac{x^{3} \cos (x) - 2 x \cos (x) - 3 \sin (x) x^{2} - \sin (x) \cdot - 2}{{(x^{3} - 2 x)}^{2}}$

Step 4.3.1.2

Multiply $- 2$ by $- 1$ .

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

Step 4.3.2

Reorder factors in $x^{3} \cos (x) - 2 x \cos (x) - 3 \sin (x) x^{2} + 2 \sin (x)$ .

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