Calculus Examples

$f (x) = 2 \cos (\frac{x}{2})$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$2 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{x}{2}])$

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $2$ .

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

Step 1.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 1.3.3

Simplify terms.

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Step 1.3.3.1

Combine $\frac{1}{2}$ and $- 2$ .

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Cancel the common factor of $- 2$ and $2$ .

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Step 1.3.3.3.1

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

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

Step 1.3.3.3.2

Cancel the common factors.

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Step 1.3.3.3.2.1

Factor $2$ out of $2$ .

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

Step 1.3.3.3.2.2

Cancel the common factor.

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

Step 1.3.3.3.2.3

Rewrite the expression.

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

Step 1.3.3.3.2.4

Divide $- \sin (\frac{x}{2})$ by $1$ .

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

Step 1.3.4

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

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

Step 1.3.5

Multiply $- 1$ by $1$ .

$f' (x) = - \sin (\frac{x}{2})$

Step 2

Find the second derivative.

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (\frac{x}{2})$ with respect to $x$ is $- \frac{d}{d x} [\sin (\frac{x}{2})]$ .

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{\cos (\frac{x}{2})}{2}$

Step 3

Find the third derivative.

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Step 3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{\cos (\frac{x}{2})}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [\cos (\frac{x}{2})]$ .

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

Combine fractions.

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Step 3.3.2.1

Multiply $\frac{1}{2}$ by $1$ .

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

Step 3.3.2.2

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

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

Step 3.3.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.3.4

Combine fractions.

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Step 3.3.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (\frac{x}{2})}{2}$ .

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

Step 3.3.4.2

Multiply $2$ by $2$ .

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

Step 3.3.5

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

$\frac{\sin (\frac{x}{2})}{4} \cdot 1$

Step 3.3.6

Multiply $\frac{\sin (\frac{x}{2})}{4}$ by $1$ .

$f''' (x) = \frac{\sin (\frac{x}{2})}{4}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $\frac{\sin (\frac{x}{2})}{4}$ .

$\frac{\sin (\frac{x}{2})}{4}$