Calculus Examples

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

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 $g (x) = x^{2}$ .

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

Step 2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3

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

$\frac{x^{2} \cos (x) - \sin (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$\frac{x^{2} \cos (x) - \sin (x) (2 x)}{x^{4}}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} \cos (x) - 2 \sin (x) x}{x^{4}}$

Step 4.2.2

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

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Step 4.2.2.1

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

$\frac{x (x \cos (x)) - 2 \sin (x) x}{x^{4}}$

Step 4.2.2.2

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

$\frac{x (x \cos (x)) + x (- 2 \sin (x))}{x^{4}}$

Step 4.2.2.3

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

$\frac{x (x \cos (x) - 2 \sin (x))}{x^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x$ out of $x^{4}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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