Calculus Examples

$y = 2 \sin (v^{2}) \cos (6 v)$

Step 1

Since $2$ is constant with respect to $v$ , the derivative of $2 \sin (v^{2}) \cos (6 v)$ with respect to $v$ is $2 \frac{d}{d v} [\sin (v^{2}) \cos (6 v)]$ .

$2 \frac{d}{d v} [\sin (v^{2}) \cos (6 v)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d v} [f (v) g (v)]$ is $f (v) \frac{d}{d v} [g (v)] + g (v) \frac{d}{d v} [f (v)]$ where $f (v) = \sin (v^{2})$ and $g (v) = \cos (6 v)$ .

$2 (\sin (v^{2}) \frac{d}{d v} [\cos (6 v)] + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = \cos (v)$ and $g (v) = 6 v$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $6 v$ .

$2 (\sin (v^{2}) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d v} [6 v]) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$2 (\sin (v^{2}) (- \sin (u_{1}) \frac{d}{d v} [6 v]) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 3.3

Replace all occurrences of $u_{1}$ with $6 v$ .

$2 (\sin (v^{2}) (- \sin (6 v) \frac{d}{d v} [6 v]) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 4

Differentiate.

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Step 4.1

Since $6$ is constant with respect to $v$ , the derivative of $6 v$ with respect to $v$ is $6 \frac{d}{d v} [v]$ .

$2 (\sin (v^{2}) (- \sin (6 v) (6 \frac{d}{d v} [v])) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 4.2

Multiply $6$ by $- 1$ .

$2 (\sin (v^{2}) (- 6 \sin (6 v) \frac{d}{d v} [v]) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 4.3

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

$2 (\sin (v^{2}) (- 6 \sin (6 v) \cdot 1) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 4.4

Multiply $- 6$ by $1$ .

$2 (\sin (v^{2}) (- 6 \sin (6 v)) + \cos (6 v) \frac{d}{d v} [\sin (v^{2})])$

Step 5

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

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $v^{2}$ .

$2 (\sin (v^{2}) (- 6 \sin (6 v)) + \cos (6 v) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d v} [v^{2}]))$

Step 5.2

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

$2 (\sin (v^{2}) (- 6 \sin (6 v)) + \cos (6 v) (\cos (u_{2}) \frac{d}{d v} [v^{2}]))$

Step 5.3

Replace all occurrences of $u_{2}$ with $v^{2}$ .

$2 (\sin (v^{2}) (- 6 \sin (6 v)) + \cos (6 v) (\cos (v^{2}) \frac{d}{d v} [v^{2}]))$

Step 6

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

$2 (\sin (v^{2}) (- 6 \sin (6 v)) + \cos (6 v) \cos (v^{2}) (2 v))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (\sin (v^{2}) (- 6 \sin (6 v))) + 2 (\cos (6 v) \cos (v^{2}) (2 v))$

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $- 6$ by $2$ .

$- 12 (\sin (v^{2}) (\sin (6 v))) + 2 (\cos (6 v) \cos (v^{2}) (2 v))$

Step 7.2.2

Multiply $2$ by $2$ .

$- 12 \sin (v^{2}) \sin (6 v) + 4 \cos (6 v) \cos (v^{2}) v$

Step 7.3

Reorder terms.

$- 12 \sin (v^{2}) \sin (6 v) + 4 v \cos (v^{2}) \cos (6 v)$