Calculus Examples

$2 \sin^{4} (2 x) - 3 \sin^{3} (4 x)$

Step 1

By the Sum Rule, the derivative of $2 \sin^{4} (2 x) - 3 \sin^{3} (4 x)$ with respect to $x$ is $\frac{d}{d x} [2 \sin^{4} (2 x)] + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$ .

$\frac{d}{d x} [2 \sin^{4} (2 x)] + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2

Evaluate $\frac{d}{d x} [2 \sin^{4} (2 x)]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin^{4} (2 x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin^{4} (2 x)]$ .

$2 \frac{d}{d x} [\sin^{4} (2 x)] + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \sin (2 x)$ .

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Step 2.2.1

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

$2 (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$2 (4 {u_{1}}^{3} \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.2.3

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

$2 (4 \sin^{3} (2 x) \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

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 $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 2.3.1

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

$2 (4 \sin^{3} (2 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.3.2

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

$2 (4 \sin^{3} (2 x) (\cos (u_{2}) \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$2 (4 \sin^{3} (2 x) (\cos (2 x) \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.4

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

$2 (4 \sin^{3} (2 x) (\cos (2 x) (2 \frac{d}{d x} [x]))) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.5

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

$2 (4 \sin^{3} (2 x) (\cos (2 x) (2 \cdot 1))) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.6

Multiply $2$ by $1$ .

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

Step 2.7

Move $2$ to the left of $\cos (2 x)$ .

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

Step 2.8

Multiply $2$ by $4$ .

$2 (8 \sin^{3} (2 x) \cos (2 x)) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 2.9

Multiply $8$ by $2$ .

$16 \sin^{3} (2 x) \cos (2 x) + \frac{d}{d x} [- 3 \sin^{3} (4 x)]$

Step 3

Evaluate $\frac{d}{d x} [- 3 \sin^{3} (4 x)]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \sin^{3} (4 x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\sin^{3} (4 x)]$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 \frac{d}{d x} [\sin^{3} (4 x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sin (4 x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{3}$ as $\sin (4 x)$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (\frac{d}{d u_{3}} [{u_{3}}^{3}] \frac{d}{d x} [\sin (4 x)])$

Step 3.2.2

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

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 {u_{3}}^{2} \frac{d}{d x} [\sin (4 x)])$

Step 3.2.3

Replace all occurrences of $u_{3}$ with $\sin (4 x)$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) \frac{d}{d x} [\sin (4 x)])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{4}$ as $4 x$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\frac{d}{d u_{4}} [\sin (u_{4})] \frac{d}{d x} [4 x]))$

Step 3.3.2

The derivative of $\sin (u_{4})$ with respect to $u_{4}$ is $\cos (u_{4})$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\cos (u_{4}) \frac{d}{d x} [4 x]))$

Step 3.3.3

Replace all occurrences of $u_{4}$ with $4 x$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\cos (4 x) \frac{d}{d x} [4 x]))$

Step 3.4

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\cos (4 x) (4 \frac{d}{d x} [x])))$

Step 3.5

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

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\cos (4 x) (4 \cdot 1)))$

Step 3.6

Multiply $4$ by $1$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (\cos (4 x) \cdot 4))$

Step 3.7

Move $4$ to the left of $\cos (4 x)$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (3 \sin^{2} (4 x) (4 \cdot \cos (4 x)))$

Step 3.8

Multiply $4$ by $3$ .

$16 \sin^{3} (2 x) \cos (2 x) - 3 (12 \sin^{2} (4 x) \cos (4 x))$

Step 3.9

Multiply $12$ by $- 3$ .

$16 \sin^{3} (2 x) \cos (2 x) - 36 \sin^{2} (4 x) \cos (4 x)$