Calculus Examples

$y = \frac{1 - \cos (4 x)}{\sin (4 x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - \cos (4 x)$ and $g (x) = \sin (4 x)$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - \cos (4 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (4 x)]$ .

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

Step 2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\sin (4 x) (0 + \frac{d}{d x} [- \cos (4 x)]) - (1 - \cos (4 x)) \frac{d}{d x} [\sin (4 x)]}{\sin^{2} (4 x)}$

Step 2.3

Add $0$ and $\frac{d}{d x} [- \cos (4 x)]$ .

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

Step 2.4

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

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

Step 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 $g (x) = 4 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\sin (4 x)$ by $1$ .

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

Step 5

Raise $\sin (4 x)$ to the power of $1$ .

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

Step 6

Raise $\sin (4 x)$ to the power of $1$ .

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

Step 7

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

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

Step 8

Differentiate.

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Step 8.1

Add $1$ and $1$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Simplify the expression.

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Step 8.4.1

Multiply $4$ by $1$ .

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

Step 8.4.2

Move $4$ to the left of $\sin^{2} (4 x)$ .

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

Step 9

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 9.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

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

Step 9.2

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

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

Step 9.3

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 10

Differentiate.

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Step 10.1

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

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

Step 10.2

Multiply $4$ by $- 1$ .

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

Step 10.3

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

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

Step 10.4

Multiply $- 4$ by $1$ .

$\frac{4 \sin^{2} (4 x) - 4 (1 - \cos (4 x)) \cos (4 x)}{\sin^{2} (4 x)}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Simplify the numerator.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Multiply $- 4$ by $1$ .

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

Step 11.3.1.2

Multiply $- 1$ by $- 4$ .

$\frac{4 \sin^{2} (4 x) - 4 \cos (4 x) + 4 \cos (4 x) \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.3.1.3

Multiply $4 \cos (4 x) \cos (4 x)$ .

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Step 11.3.1.3.1

Raise $\cos (4 x)$ to the power of $1$ .

$\frac{4 \sin^{2} (4 x) - 4 \cos (4 x) + 4 (\cos^{1} (4 x) \cos (4 x))}{\sin^{2} (4 x)}$

Step 11.3.1.3.2

Raise $\cos (4 x)$ to the power of $1$ .

$\frac{4 \sin^{2} (4 x) - 4 \cos (4 x) + 4 (\cos^{1} (4 x) \cos^{1} (4 x))}{\sin^{2} (4 x)}$

Step 11.3.1.3.3

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

$\frac{4 \sin^{2} (4 x) - 4 \cos (4 x) + 4 {\cos (4 x)}^{1 + 1}}{\sin^{2} (4 x)}$

Step 11.3.1.3.4

Add $1$ and $1$ .

$\frac{4 \sin^{2} (4 x) - 4 \cos (4 x) + 4 \cos^{2} (4 x)}{\sin^{2} (4 x)}$

Step 11.3.2

Move $4 \cos^{2} (4 x)$ .

$\frac{4 \sin^{2} (4 x) + 4 \cos^{2} (4 x) - 4 \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.3.3

Factor $4$ out of $4 \sin^{2} (4 x)$ .

$\frac{4 (\sin^{2} (4 x)) + 4 \cos^{2} (4 x) - 4 \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.3.4

Factor $4$ out of $4 \cos^{2} (4 x)$ .

$\frac{4 (\sin^{2} (4 x)) + 4 (\cos^{2} (4 x)) - 4 \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.3.5

Factor $4$ out of $4 (\sin^{2} (4 x)) + 4 (\cos^{2} (4 x))$ .

$\frac{4 (\sin^{2} (4 x) + \cos^{2} (4 x)) - 4 \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.3.6

Apply pythagorean identity.

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

Step 11.3.7

Multiply $4$ by $1$ .

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

Step 11.4

Factor $4$ out of $4 - 4 \cos (4 x)$ .

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Step 11.4.1

Factor $4$ out of $4$ .

$\frac{4 (1) - 4 \cos (4 x)}{\sin^{2} (4 x)}$

Step 11.4.2

Factor $4$ out of $- 4 \cos (4 x)$ .

$\frac{4 (1) + 4 (- \cos (4 x))}{\sin^{2} (4 x)}$

Step 11.4.3

Factor $4$ out of $4 (1) + 4 (- \cos (4 x))$ .

$\frac{4 (1 - \cos (4 x))}{\sin^{2} (4 x)}$