Calculus Examples

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

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 $f (x) = 1 - \cos (x)$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

$\frac{x^{2} \frac{d}{d x} [1 - \cos (x)] - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [1 - \cos (x)] - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2

By the Sum Rule, the derivative of $1 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{x^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{2} (0 + \frac{d}{d x} [- \cos (x)]) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

Add $0$ and $\frac{d}{d x} [- \cos (x)]$ .

$\frac{x^{2} \frac{d}{d x} [- \cos (x)] - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$\frac{x^{2} (- \frac{d}{d x} [\cos (x)]) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3

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

$\frac{x^{2} (- - \sin (x)) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} (1 \sin (x)) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

Multiply $\sin (x)$ by $1$ .

$\frac{x^{2} \sin (x) - (1 - \cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3

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

$\frac{x^{2} \sin (x) - (1 - \cos (x)) (2 x)}{x^{4}}$

Step 4.4

Simplify with factoring out.

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Step 4.4.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} \sin (x) - 2 (1 - \cos (x)) x}{x^{4}}$

Step 4.4.2

Factor $x$ out of $x^{2} \sin (x) - 2 (1 - \cos (x)) x$ .

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Step 4.4.2.1

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

$\frac{x (x \sin (x)) - 2 (1 - \cos (x)) x}{x^{4}}$

Step 4.4.2.2

Factor $x$ out of $- 2 (1 - \cos (x)) x$ .

$\frac{x (x \sin (x)) + x (- 2 (1 - \cos (x)))}{x^{4}}$

Step 4.4.2.3

Factor $x$ out of $x (x \sin (x)) + x (- 2 (1 - \cos (x)))$ .

$\frac{x (x \sin (x) - 2 (1 - \cos (x)))}{x^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $- 2$ by $1$ .

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

Step 6.2.2

Multiply $- 1$ by $- 2$ .

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