Calculus Examples

\sin^{2} (6 x)

$\sin^{2} (6 x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = \sin (6 x)

$g (x) = \sin (6 x)$ .

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Step 1.1

To apply the Chain Rule, set

u_{1}

$u_{1}$ as

\sin (6 x)

$\sin (6 x)$ .

\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (6 x)]

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

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d u_{1}} [{u_{1}}^{n}]

$\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is

n {u_{1}}^{n - 1}

$n {u_{1}}^{n - 1}$ where

n = 2

$n = 2$ .

2 u_{1} \frac{d}{d x} [\sin (6 x)]

$2 u_{1} \frac{d}{d x} [\sin (6 x)]$

Step 1.3

Replace all occurrences of

u_{1}

$u_{1}$ with

\sin (6 x)

$\sin (6 x)$ .

2 \sin (6 x) \frac{d}{d x} [\sin (6 x)]

$2 \sin (6 x) \frac{d}{d x} [\sin (6 x)]$

2 \sin (6 x) \frac{d}{d x} [\sin (6 x)]

$2 \sin (6 x) \frac{d}{d x} [\sin (6 x)]$

Step 2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = \sin (x)

$f (x) = \sin (x)$ and

g (x) = 6 x

$g (x) = 6 x$ .

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Step 2.1

To apply the Chain Rule, set

u_{2}

$u_{2}$ as

6 x

$6 x$ .

2 \sin (6 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [6 x])

$2 \sin (6 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [6 x])$

Step 2.2

The derivative of

\sin (u_{2})

$\sin (u_{2})$ with respect to

u_{2}

$u_{2}$ is

\cos (u_{2})

$\cos (u_{2})$ .

2 \sin (6 x) (\cos (u_{2}) \frac{d}{d x} [6 x])

$2 \sin (6 x) (\cos (u_{2}) \frac{d}{d x} [6 x])$

Step 2.3

Replace all occurrences of

u_{2}

$u_{2}$ with

6 x

$6 x$ .

2 \sin (6 x) (\cos (6 x) \frac{d}{d x} [6 x])

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

2 \sin (6 x) (\cos (6 x) \frac{d}{d x} [6 x])

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

Step 3

Differentiate.

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Step 3.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

6 x

$6 x$ with respect to

x

$x$ is

6 \frac{d}{d x} [x]

$6 \frac{d}{d x} [x]$ .

2 \sin (6 x) (\cos (6 x) (6 \frac{d}{d x} [x]))

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

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

2 \sin (6 x) (\cos (6 x) (6 \cdot 1))

$2 \sin (6 x) (\cos (6 x) (6 \cdot 1))$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply

6

$6$ by

1

$1$ .

2 \sin (6 x) (\cos (6 x) \cdot 6)

$2 \sin (6 x) (\cos (6 x) \cdot 6)$

Step 3.3.2

Move

6

$6$ to the left of

\cos (6 x)

$\cos (6 x)$ .

2 \sin (6 x) (6 \cdot \cos (6 x))

$2 \sin (6 x) (6 \cdot \cos (6 x))$

2 \sin (6 x) (6 \cdot \cos (6 x))

$2 \sin (6 x) (6 \cdot \cos (6 x))$

2 \sin (6 x) (6 \cdot \cos (6 x))

$2 \sin (6 x) (6 \cdot \cos (6 x))$

Step 4

Simplify.

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Step 4.1

Multiply

6

$6$ by

2

$2$ .

12 \sin (6 x) \cos (6 x)

$12 \sin (6 x) \cos (6 x)$

Step 4.2

Reorder the factors of

12 \sin (6 x) \cos (6 x)

$12 \sin (6 x) \cos (6 x)$ .

12 \cos (6 x) \sin (6 x)

$12 \cos (6 x) \sin (6 x)$

12 \cos (6 x) \sin (6 x)

$12 \cos (6 x) \sin (6 x)$

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