Calculus Examples

$1.8 \sin^{3} (\frac{π}{12} x - \frac{π}{2}) + 98.6$

Step 1

By the Sum Rule, the derivative of $1.8 \sin^{3} (\frac{π}{12} x - \frac{π}{2}) + 98.6$ with respect to $x$ is $\frac{d}{d x} [1.8 \sin^{3} (\frac{π}{12} x - \frac{π}{2})] + \frac{d}{d x} [98.6]$ .

$\frac{d}{d x} [1.8 \sin^{3} (\frac{π}{12} x - \frac{π}{2})] + \frac{d}{d x} [98.6]$

Step 2

Evaluate $\frac{d}{d x} [1.8 \sin^{3} (\frac{π}{12} x - \frac{π}{2})]$ .

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Step 2.1

Combine $\frac{π}{12}$ and $x$ .

$\frac{d}{d x} [1.8 \sin^{3} (\frac{π x}{12} - \frac{π}{2})] + \frac{d}{d x} [98.6]$

Step 2.2

Since $1.8$ is constant with respect to $x$ , the derivative of $1.8 \sin^{3} (\frac{π x}{12} - \frac{π}{2})$ with respect to $x$ is $1.8 \frac{d}{d x} [\sin^{3} (\frac{π x}{12} - \frac{π}{2})]$ .

$1.8 \frac{d}{d x} [\sin^{3} (\frac{π x}{12} - \frac{π}{2})] + \frac{d}{d x} [98.6]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sin (\frac{π x}{12} - \frac{π}{2})$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $\sin (\frac{π x}{12} - \frac{π}{2})$ .

$1.8 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\sin (\frac{π x}{12} - \frac{π}{2})]) + \frac{d}{d x} [98.6]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$1.8 (3 {u_{1}}^{2} \frac{d}{d x} [\sin (\frac{π x}{12} - \frac{π}{2})]) + \frac{d}{d x} [98.6]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $\sin (\frac{π x}{12} - \frac{π}{2})$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) \frac{d}{d x} [\sin (\frac{π x}{12} - \frac{π}{2})]) + \frac{d}{d x} [98.6]$

Step 2.4

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 $g (x) = \frac{π x}{12} - \frac{π}{2}$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $\frac{π x}{12} - \frac{π}{2}$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [\frac{π x}{12} - \frac{π}{2}])) + \frac{d}{d x} [98.6]$

Step 2.4.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (u_{2}) \frac{d}{d x} [\frac{π x}{12} - \frac{π}{2}])) + \frac{d}{d x} [98.6]$

Step 2.4.3

Replace all occurrences of $u_{2}$ with $\frac{π x}{12} - \frac{π}{2}$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) \frac{d}{d x} [\frac{π x}{12} - \frac{π}{2}])) + \frac{d}{d x} [98.6]$

Step 2.5

By the Sum Rule, the derivative of $\frac{π x}{12} - \frac{π}{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [- \frac{π}{2}]$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) (\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [98.6]$

Step 2.6

Since $\frac{π}{12}$ is constant with respect to $x$ , the derivative of $\frac{π x}{12}$ with respect to $x$ is $\frac{π}{12} \frac{d}{d x} [x]$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) (\frac{π}{12} \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [98.6]$

Step 2.7

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

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) (\frac{π}{12} \cdot 1 + \frac{d}{d x} [- \frac{π}{2}]))) + \frac{d}{d x} [98.6]$

Step 2.8

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

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) (\frac{π}{12} \cdot 1 + 0))) + \frac{d}{d x} [98.6]$

Step 2.9

Multiply $\frac{π}{12}$ by $1$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) (\frac{π}{12} + 0))) + \frac{d}{d x} [98.6]$

Step 2.10

Add $\frac{π}{12}$ and $0$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) (\cos (\frac{π x}{12} - \frac{π}{2}) \frac{π}{12})) + \frac{d}{d x} [98.6]$

Step 2.11

Combine $\cos (\frac{π x}{12} - \frac{π}{2})$ and $\frac{π}{12}$ .

$1.8 (3 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) \frac{\cos (\frac{π x}{12} - \frac{π}{2}) π}{12}) + \frac{d}{d x} [98.6]$

Step 2.12

Combine $\frac{\cos (\frac{π x}{12} - \frac{π}{2}) π}{12}$ and $3$ .

$1.8 (\frac{\cos (\frac{π x}{12} - \frac{π}{2}) π \cdot 3}{12} \sin^{2} (\frac{π x}{12} - \frac{π}{2})) + \frac{d}{d x} [98.6]$

Step 2.13

Combine $\frac{\cos (\frac{π x}{12} - \frac{π}{2}) π \cdot 3}{12}$ and $\sin^{2} (\frac{π x}{12} - \frac{π}{2})$ .

$1.8 \frac{\cos (\frac{π x}{12} - \frac{π}{2}) π \cdot 3 \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{12} + \frac{d}{d x} [98.6]$

Step 2.14

Move $3$ to the left of $\cos (\frac{π x}{12} - \frac{π}{2}) π$ .

$1.8 \frac{3 \cdot (\cos (\frac{π x}{12} - \frac{π}{2}) π) \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{12} + \frac{d}{d x} [98.6]$

Step 2.15

Cancel the common factor of $3$ and $12$ .

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Step 2.15.1

Factor $3$ out of $3 \cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2})$ .

$1.8 \frac{3 (\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2}))}{12} + \frac{d}{d x} [98.6]$

Step 2.15.2

Cancel the common factors.

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Step 2.15.2.1

Factor $3$ out of $12$ .

$1.8 \frac{3 (\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2}))}{3 \cdot 4} + \frac{d}{d x} [98.6]$

Step 2.15.2.2

Cancel the common factor.

$1.8 \frac{3 (\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2}))}{3 \cdot 4} + \frac{d}{d x} [98.6]$

Step 2.15.2.3

Rewrite the expression.

$1.8 \frac{\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4} + \frac{d}{d x} [98.6]$

Step 2.16

Combine $1.8$ and $\frac{\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4}$ .

$\frac{1.8 (\cos (\frac{π x}{12} - \frac{π}{2}) π \sin^{2} (\frac{π x}{12} - \frac{π}{2}))}{4} + \frac{d}{d x} [98.6]$

Step 2.17

Multiply $π$ by $1.8$ .

$\frac{5.65486677 \cos (\frac{π x}{12} - \frac{π}{2}) \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4} + \frac{d}{d x} [98.6]$

Step 3

Since $98.6$ is constant with respect to $x$ , the derivative of $98.6$ with respect to $x$ is $0$ .

$\frac{5.65486677 \cos (\frac{π x}{12} - \frac{π}{2}) \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4} + 0$

Step 4

Simplify.

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Step 4.1

Add $\frac{5.65486677 \cos (\frac{π x}{12} - \frac{π}{2}) \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4}$ and $0$ .

$\frac{5.65486677 \cos (\frac{π x}{12} - \frac{π}{2}) \sin^{2} (\frac{π x}{12} - \frac{π}{2})}{4}$

Step 4.2

Reorder terms.

$\frac{5.65486677 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})}{4}$

Step 4.3

Factor $5.65486677$ out of $5.65486677 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})$ .

$\frac{5.65486677 (\sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2}))}{4}$

Step 4.4

Factor $4$ out of $4$ .

$\frac{5.65486677 (\sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2}))}{4 (1)}$

Step 4.5

Separate fractions.

$\frac{5.65486677}{4} \cdot \frac{\sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})}{1}$

Step 4.6

Divide $5.65486677$ by $4$ .

$1.41371669 \frac{\sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})}{1}$

Step 4.7

Divide $\sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})$ by $1$ .

$1.41371669 \sin^{2} (\frac{π x}{12} - \frac{π}{2}) \cos (\frac{π x}{12} - \frac{π}{2})$