Calculus Examples

$f (x) = 5 \cos (\frac{1}{2} x^{2} - \frac{3}{4} \cdot π)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $π$ and $\frac{3}{4}$ .

$\frac{d}{d x} [5 \cos (\frac{1}{2} x^{2} - \frac{π \cdot 3}{4})]$

Step 1.2

Combine fractions.

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Step 1.2.1

Move $3$ to the left of $π$ .

$\frac{d}{d x} [5 \cos (\frac{1}{2} x^{2} - \frac{3 \cdot π}{4})]$

Step 1.2.2

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{d}{d x} [5 \cos (\frac{x^{2}}{2} - \frac{3 π}{4})]$

Step 1.3

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (\frac{x^{2}}{2} - \frac{3 π}{4})$ with respect to $x$ is $5 \frac{d}{d x} [\cos (\frac{x^{2}}{2} - \frac{3 π}{4})]$ .

$5 \frac{d}{d x} [\cos (\frac{x^{2}}{2} - \frac{3 π}{4})]$

Step 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 $g (x) = \frac{x^{2}}{2} - \frac{3 π}{4}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x^{2}}{2} - \frac{3 π}{4}$ .

$5 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{x^{2}}{2} - \frac{3 π}{4}])$

Step 2.2

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

$5 (- \sin (u) \frac{d}{d x} [\frac{x^{2}}{2} - \frac{3 π}{4}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{x^{2}}{2} - \frac{3 π}{4}$ .

$5 (- \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) \frac{d}{d x} [\frac{x^{2}}{2} - \frac{3 π}{4}])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $5$ .

$- 5 (\sin (\frac{x^{2}}{2} - \frac{3 π}{4}) \frac{d}{d x} [\frac{x^{2}}{2} - \frac{3 π}{4}])$

Step 3.2

By the Sum Rule, the derivative of $\frac{x^{2}}{2} - \frac{3 π}{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [- \frac{3 π}{4}]$ .

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.3

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

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.4

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

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{1}{2} (2 x) + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.5

Simplify terms.

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Step 3.5.1

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

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{2}{2} x + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.5.2

Combine $\frac{2}{2}$ and $x$ .

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{2 x}{2} + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.5.3

Cancel the common factor of $2$ .

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Step 3.5.3.1

Cancel the common factor.

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (\frac{2 x}{2} + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.5.3.2

Divide $x$ by $1$ .

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (x + \frac{d}{d x} [- \frac{3 π}{4}])$

Step 3.6

Since $- \frac{3 π}{4}$ is constant with respect to $x$ , the derivative of $- \frac{3 π}{4}$ with respect to $x$ is $0$ .

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) (x + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $x$ and $0$ .

$- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) x$

Step 3.7.2

Reorder the factors of $- 5 \sin (\frac{x^{2}}{2} - \frac{3 π}{4}) x$ .

$- 5 x \sin (\frac{x^{2}}{2} - \frac{3 π}{4})$