Calculus Examples

$y = 12 \sin (w t + π)$

Step 1

Find the first derivative.

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Step 1.1

Since $12$ is constant with respect to $w$ , the derivative of $12 \sin (w t + π)$ with respect to $w$ is $12 \frac{d}{d w} [\sin (w t + π)]$ .

$12 \frac{d}{d w} [\sin (w t + π)]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \sin (w)$ and $g (w) = w t + π$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $w t + π$ .

$12 (\frac{d}{d u} [\sin (u)] \frac{d}{d w} [w t + π])$

Step 1.2.2

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

$12 (\cos (u) \frac{d}{d w} [w t + π])$

Step 1.2.3

Replace all occurrences of $u$ with $w t + π$ .

$12 (\cos (w t + π) \frac{d}{d w} [w t + π])$

Step 1.3

Differentiate.

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Step 1.3.1

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

$12 \cos (w t + π) (\frac{d}{d w} [w t] + \frac{d}{d w} [π])$

Step 1.3.2

Since $t$ is constant with respect to $w$ , the derivative of $w t$ with respect to $w$ is $t \frac{d}{d w} [w]$ .

$12 \cos (w t + π) (t \frac{d}{d w} [w] + \frac{d}{d w} [π])$

Step 1.3.3

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

$12 \cos (w t + π) (t \cdot 1 + \frac{d}{d w} [π])$

Step 1.3.4

Multiply $t$ by $1$ .

$12 \cos (w t + π) (t + \frac{d}{d w} [π])$

Step 1.3.5

Since $π$ is constant with respect to $w$ , the derivative of $π$ with respect to $w$ is $0$ .

$12 \cos (w t + π) (t + 0)$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $t$ and $0$ .

$12 \cos (w t + π) t$

Step 1.3.6.2

Reorder the factors of $12 \cos (w t + π) t$ .

$f' (w) = 12 t \cos (w t + π)$

Step 2

Find the second derivative.

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Step 2.1

Since $12 t$ is constant with respect to $w$ , the derivative of $12 t \cos (w t + π)$ with respect to $w$ is $12 t \frac{d}{d w} [\cos (w t + π)]$ .

$12 t \frac{d}{d w} [\cos (w t + π)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \cos (w)$ and $g (w) = w t + π$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $w t + π$ .

$12 t (\frac{d}{d u} [\cos (u)] \frac{d}{d w} [w t + π])$

Step 2.2.2

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

$12 t (- \sin (u) \frac{d}{d w} [w t + π])$

Step 2.2.3

Replace all occurrences of $u$ with $w t + π$ .

$12 t (- \sin (w t + π) \frac{d}{d w} [w t + π])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $12$ .

$- 12 t (\sin (w t + π) \frac{d}{d w} [w t + π])$

Step 2.3.2

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

$- 12 t (\sin (w t + π) (\frac{d}{d w} [w t] + \frac{d}{d w} [π]))$

Step 2.3.3

Since $t$ is constant with respect to $w$ , the derivative of $w t$ with respect to $w$ is $t \frac{d}{d w} [w]$ .

$- 12 t (\sin (w t + π) (t \frac{d}{d w} [w] + \frac{d}{d w} [π]))$

Step 2.3.4

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

$- 12 t (\sin (w t + π) (t \cdot 1 + \frac{d}{d w} [π]))$

Step 2.3.5

Multiply $t$ by $1$ .

$- 12 t (\sin (w t + π) (t + \frac{d}{d w} [π]))$

Step 2.3.6

Since $π$ is constant with respect to $w$ , the derivative of $π$ with respect to $w$ is $0$ .

$- 12 t (\sin (w t + π) (t + 0))$

Step 2.3.7

Add $t$ and $0$ .

$- 12 t (\sin (w t + π) t)$

Step 2.4

Raise $t$ to the power of $1$ .

$- 12 (t^{1} t) \sin (w t + π)$

Step 2.5

Raise $t$ to the power of $1$ .

$- 12 (t^{1} t^{1}) \sin (w t + π)$

Step 2.6

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

$- 12 t^{1 + 1} \sin (w t + π)$

Step 2.7

Add $1$ and $1$ .

$f'' (w) = - 12 t^{2} \sin (w t + π)$

Step 3

Find the third derivative.

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Step 3.1

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

$- 12 t^{2} \frac{d}{d w} [\sin (w t + π)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \sin (w)$ and $g (w) = w t + π$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $w t + π$ .

$- 12 t^{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d w} [w t + π])$

Step 3.2.2

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

$- 12 t^{2} (\cos (u) \frac{d}{d w} [w t + π])$

Step 3.2.3

Replace all occurrences of $u$ with $w t + π$ .

$- 12 t^{2} (\cos (w t + π) \frac{d}{d w} [w t + π])$

Step 3.3

Differentiate.

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Step 3.3.1

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

$- 12 t^{2} (\cos (w t + π) (\frac{d}{d w} [w t] + \frac{d}{d w} [π]))$

Step 3.3.2

Since $t$ is constant with respect to $w$ , the derivative of $w t$ with respect to $w$ is $t \frac{d}{d w} [w]$ .

$- 12 t^{2} (\cos (w t + π) (t \frac{d}{d w} [w] + \frac{d}{d w} [π]))$

Step 3.3.3

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

$- 12 t^{2} (\cos (w t + π) (t \cdot 1 + \frac{d}{d w} [π]))$

Step 3.3.4

Multiply $t$ by $1$ .

$- 12 t^{2} (\cos (w t + π) (t + \frac{d}{d w} [π]))$

Step 3.3.5

Since $π$ is constant with respect to $w$ , the derivative of $π$ with respect to $w$ is $0$ .

$- 12 t^{2} (\cos (w t + π) (t + 0))$

Step 3.3.6

Add $t$ and $0$ .

$- 12 t^{2} (\cos (w t + π) t)$

Step 3.4

Raise $t$ to the power of $1$ .

$- 12 (t^{1} t^{2}) \cos (w t + π)$

Step 3.5

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

$- 12 t^{1 + 2} \cos (w t + π)$

Step 3.6

Add $1$ and $2$ .

$f''' (w) = - 12 t^{3} \cos (w t + π)$