Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{π}{2} (x - 1)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{π}{2} (x - 1)$ .

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

Step 2.2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Move $2$ to the left of $\cos (\frac{π}{2} (x - 1))$ .

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

Step 3.3

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

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

Step 3.4

Simplify terms.

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Step 3.4.1

Multiply $\frac{π}{2}$ by $\frac{2 \cos (\frac{π}{2} (x - 1))}{π}$ .

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

Step 3.4.2

Move $2$ to the left of $π$ .

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

Step 3.4.3

Cancel the common factor of $2$ .

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Step 3.4.3.1

Cancel the common factor.

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

Step 3.4.3.2

Rewrite the expression.

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

Step 3.4.4

Cancel the common factor of $π$ .

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Step 3.4.4.1

Cancel the common factor.

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

Step 3.4.4.2

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

$\cos (\frac{π}{2} (x - 1)) (1 + 0)$

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $1$ and $0$ .

$\cos (\frac{π}{2} (x - 1)) \cdot 1$

Step 3.8.2

Multiply $\cos (\frac{π}{2} (x - 1))$ by $1$ .

$\cos (\frac{π}{2} (x - 1))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Combine $\frac{π}{2}$ and $- 1$ .

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

Step 4.2.2

Move $- 1$ to the left of $π$ .

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

Step 4.2.3

Rewrite $- 1 π$ as $- π$ .

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

Step 4.2.4

Move the negative in front of the fraction.

$\cos (\frac{π}{2} x - \frac{π}{2})$