Calculus Examples

$f (x) = \cos (\frac{π x}{2})$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $- 1$ by $1$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

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 $g (x) = \frac{π x}{2}$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Differentiate using the Constant Multiple Rule.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $\frac{π}{2}$ by $\frac{\cos (\frac{π x}{2}) π}{2}$ .

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

Step 2.4

Raise $π$ to the power of $1$ .

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

Step 2.5

Raise $π$ to the power of $1$ .

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

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.7

Simplify the expression.

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Step 2.7.1

Add $1$ and $1$ .

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

Step 2.7.2

Multiply $2$ by $2$ .

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

Step 2.8

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

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

Step 2.9

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{π^{2} \cos (\frac{π x}{2})}{4}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{π^{2} \cos (\frac{π x}{2})}{4}$ .

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