Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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 $g (x) = \sin (3 x)$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (3 x)$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\sin (3 x)$ .

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

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 $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $3$ by $1$ .

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

Step 2.6

Move $3$ to the left of $\cos (3 x)$ .

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

Step 2.7

Multiply $3$ by $4$ .

$12 \sin^{3} (3 x) \cos (3 x) + \frac{d}{d x} [- \cos^{4} (3 x)]$

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos^{4} (3 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos^{4} (3 x)]$ .

$12 \sin^{3} (3 x) \cos (3 x) - \frac{d}{d x} [\cos^{4} (3 x)]$

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 $f (x) = x^{4}$ and $g (x) = \cos (3 x)$ .

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Step 3.2.1

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

$12 \sin^{3} (3 x) \cos (3 x) - (\frac{d}{d u_{3}} [{u_{3}}^{4}] \frac{d}{d x} [\cos (3 x)])$

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 $n = 4$ .

$12 \sin^{3} (3 x) \cos (3 x) - (4 {u_{3}}^{3} \frac{d}{d x} [\cos (3 x)])$

Step 3.2.3

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

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) \frac{d}{d x} [\cos (3 x)])$

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 $f (x) = \cos (x)$ and $g (x) = 3 x$ .

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Step 3.3.1

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

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) (\frac{d}{d u_{4}} [\cos (u_{4})] \frac{d}{d x} [3 x]))$

Step 3.3.2

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

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) (- \sin (u_{4}) \frac{d}{d x} [3 x]))$

Step 3.3.3

Replace all occurrences of $u_{4}$ with $3 x$ .

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) (- \sin (3 x) \frac{d}{d x} [3 x]))$

Step 3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) (- \sin (3 x) (3 \frac{d}{d x} [x])))$

Step 3.5

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

$12 \sin^{3} (3 x) \cos (3 x) - (4 \cos^{3} (3 x) (- \sin (3 x) (3 \cdot 1)))$

Step 3.6

Multiply $3$ by $1$ .

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

Step 3.7

Multiply $3$ by $- 1$ .

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

Step 3.8

Multiply $- 3$ by $4$ .

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

Step 3.9

Multiply $- 12$ by $- 1$ .

$12 \sin^{3} (3 x) \cos (3 x) + 12 \cos^{3} (3 x) \sin (3 x)$

Step 4

Simplify.

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Step 4.1

Factor $12 \sin (3 x) \cos (3 x)$ out of $12 \sin^{3} (3 x) \cos (3 x) + 12 \cos^{3} (3 x) \sin (3 x)$ .

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Step 4.1.1

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

$12 \sin (3 x) \cos (3 x) (\sin^{2} (3 x)) + 12 \cos^{3} (3 x) \sin (3 x)$

Step 4.1.2

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

$12 \sin (3 x) \cos (3 x) (\sin^{2} (3 x)) + 12 \sin (3 x) \cos (3 x) (\cos^{2} (3 x))$

Step 4.1.3

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

$12 \sin (3 x) \cos (3 x) (\sin^{2} (3 x) + \cos^{2} (3 x))$

Step 4.2

Apply pythagorean identity.

$12 \sin (3 x) \cos (3 x) \cdot 1$

Step 4.3

Multiply $12$ by $1$ .

$12 \sin (3 x) \cos (3 x)$