Calculus Examples

$L (x) = \sin (\frac{π}{6}) + \cos (\frac{π}{6}) (x - \frac{π}{6})$

Step 1

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{d}{d x} [\frac{1}{2} + \cos (\frac{π}{6}) (x - \frac{π}{6})]$

Step 2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{d}{d x} [\frac{1}{2} + \frac{\sqrt{3}}{2} (x - \frac{π}{6})]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $\frac{1}{2} + \frac{\sqrt{3}}{2} (x - \frac{π}{6})$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [\frac{\sqrt{3}}{2} (x - \frac{π}{6})]$ .

$\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [\frac{\sqrt{3}}{2} (x - \frac{π}{6})]$

Step 3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{\sqrt{3}}{2} (x - \frac{π}{6})]$

Step 4

Evaluate $\frac{d}{d x} [\frac{\sqrt{3}}{2} (x - \frac{π}{6})]$ .

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Step 4.1

Since $\frac{\sqrt{3}}{2}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt{3}}{2} (x - \frac{π}{6})$ with respect to $x$ is $\frac{\sqrt{3}}{2} \frac{d}{d x} [x - \frac{π}{6}]$ .

$0 + \frac{\sqrt{3}}{2} \frac{d}{d x} [x - \frac{π}{6}]$

Step 4.2

By the Sum Rule, the derivative of $x - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]$ .

$0 + \frac{\sqrt{3}}{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}])$

Step 4.3

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

$0 + \frac{\sqrt{3}}{2} (1 + \frac{d}{d x} [- \frac{π}{6}])$

Step 4.4

Since $- \frac{π}{6}$ is constant with respect to $x$ , the derivative of $- \frac{π}{6}$ with respect to $x$ is $0$ .

$0 + \frac{\sqrt{3}}{2} (1 + 0)$

Step 4.5

Add $1$ and $0$ .

$0 + \frac{\sqrt{3}}{2} \cdot 1$

Step 4.6

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$0 + \frac{\sqrt{3}}{2}$

Step 5

Add $0$ and $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{2}$

Decimal Form:

$0.86602540 \dots$