Calculus Examples

$y = \frac{\csc^{2} (\frac{x}{2})}{3 x + 4}$

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 $f (x) = \csc^{2} (\frac{x}{2})$ and $g (x) = 3 x + 4$ .

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

Step 2

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

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Step 2.1

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

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

Step 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 $n = 2$ .

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

Step 2.3

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

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

Step 3

Move $2$ to the left of $3 x + 4$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

The derivative of $\csc (u_{2})$ with respect to $u_{2}$ is $- \csc (u_{2}) \cot (u_{2})$ .

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

Step 4.3

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

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

Raise $\csc (\frac{x}{2})$ to the power of $1$ .

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

Step 7

Raise $\csc (\frac{x}{2})$ to the power of $1$ .

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

Step 8

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

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

Step 9

Add $1$ and $1$ .

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

Step 10

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 11

Simplify terms.

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Step 11.1

Combine $\frac{1}{2}$ and $- 2$ .

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

Step 11.2

Combine $\csc^{2} (\frac{x}{2})$ and $\frac{- 2}{2}$ .

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

Step 11.3

Combine $\cot (\frac{x}{2})$ and $\frac{\csc^{2} (\frac{x}{2}) \cdot - 2}{2}$ .

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

Step 11.4

Move $- 2$ to the left of $\csc^{2} (\frac{x}{2})$ .

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

Step 11.5

Cancel the common factor of $- 2$ and $2$ .

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Step 11.5.1

Factor $2$ out of $\cot (\frac{x}{2}) (- 2 \csc^{2} (\frac{x}{2}))$ .

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

Step 11.5.2

Cancel the common factors.

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Step 11.5.2.1

Factor $2$ out of $2$ .

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

Step 11.5.2.2

Cancel the common factor.

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

Step 11.5.2.3

Rewrite the expression.

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

Step 11.5.2.4

Divide $\cot (\frac{x}{2}) (- \csc^{2} (\frac{x}{2}))$ by $1$ .

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

Step 12

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

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

Step 13

Multiply $\cot (\frac{x}{2})$ by $1$ .

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Multiply $3$ by $1$ .

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

Step 18

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{- \csc^{2} (\frac{x}{2}) (3 x + 4) \cot (\frac{x}{2}) - \csc^{2} (\frac{x}{2}) (3 + 0)}{{(3 x + 4)}^{2}}$

Step 19

Simplify the expression.

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Step 19.1

Add $3$ and $0$ .

$\frac{- \csc^{2} (\frac{x}{2}) (3 x + 4) \cot (\frac{x}{2}) - \csc^{2} (\frac{x}{2}) \cdot 3}{{(3 x + 4)}^{2}}$

Step 19.2

Multiply $3$ by $- 1$ .

$\frac{- \csc^{2} (\frac{x}{2}) (3 x + 4) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$\frac{(- \csc^{2} (\frac{x}{2}) (3 x) - \csc^{2} (\frac{x}{2}) \cdot 4) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.2

Apply the distributive property.

$\frac{- \csc^{2} (\frac{x}{2}) (3 x) \cot (\frac{x}{2}) - \csc^{2} (\frac{x}{2}) \cdot 4 \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.3

Simplify the numerator.

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Step 20.3.1

Simplify each term.

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Step 20.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 1 \cdot 3 \csc^{2} (\frac{x}{2}) x \cot (\frac{x}{2}) - \csc^{2} (\frac{x}{2}) \cdot 4 \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.3.1.2

Multiply $- 1$ by $3$ .

$\frac{- 3 \csc^{2} (\frac{x}{2}) x \cot (\frac{x}{2}) - \csc^{2} (\frac{x}{2}) \cdot 4 \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.3.1.3

Multiply $4$ by $- 1$ .

$\frac{- 3 \csc^{2} (\frac{x}{2}) x \cot (\frac{x}{2}) - 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.3.2

Reorder factors in $- 3 \csc^{2} (\frac{x}{2}) x \cot (\frac{x}{2}) - 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})$ .

$\frac{- 3 x \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.4

Factor $\csc^{2} (\frac{x}{2})$ out of $- 3 x \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})$ .

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Step 20.4.1

Factor $\csc^{2} (\frac{x}{2})$ out of $- 3 x \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2})$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2})) - 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2}) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.4.2

Factor $\csc^{2} (\frac{x}{2})$ out of $- 4 \csc^{2} (\frac{x}{2}) \cot (\frac{x}{2})$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) (- 4 \cot (\frac{x}{2})) - 3 \csc^{2} (\frac{x}{2})}{{(3 x + 4)}^{2}}$

Step 20.4.3

Factor $\csc^{2} (\frac{x}{2})$ out of $- 3 \csc^{2} (\frac{x}{2})$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) (- 4 \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) \cdot - 3}{{(3 x + 4)}^{2}}$

Step 20.4.4

Factor $\csc^{2} (\frac{x}{2})$ out of $\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) (- 4 \cot (\frac{x}{2}))$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2}) - 4 \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) \cdot - 3}{{(3 x + 4)}^{2}}$

Step 20.4.5

Factor $\csc^{2} (\frac{x}{2})$ out of $\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2}) - 4 \cot (\frac{x}{2})) + \csc^{2} (\frac{x}{2}) \cdot - 3$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 3 x \cot (\frac{x}{2}) - 4 \cot (\frac{x}{2}) - 3)}{{(3 x + 4)}^{2}}$

Step 20.5

Factor $- 1$ out of $- 3 x \cot (\frac{x}{2})$ .

$\frac{\csc^{2} (\frac{x}{2}) (- (3 x \cot (\frac{x}{2})) - 4 \cot (\frac{x}{2}) - 3)}{{(3 x + 4)}^{2}}$

Step 20.6

Factor $- 1$ out of $- 4 \cot (\frac{x}{2})$ .

$\frac{\csc^{2} (\frac{x}{2}) (- (3 x \cot (\frac{x}{2})) - (4 \cot (\frac{x}{2})) - 3)}{{(3 x + 4)}^{2}}$

Step 20.7

Factor $- 1$ out of $- (3 x \cot (\frac{x}{2})) - (4 \cot (\frac{x}{2}))$ .

$\frac{\csc^{2} (\frac{x}{2}) (- (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2})) - 3)}{{(3 x + 4)}^{2}}$

Step 20.8

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{\csc^{2} (\frac{x}{2}) (- (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2})) - 1 (3))}{{(3 x + 4)}^{2}}$

Step 20.9

Factor $- 1$ out of $- (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2})) - 1 (3)$ .

$\frac{\csc^{2} (\frac{x}{2}) (- (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2}) + 3))}{{(3 x + 4)}^{2}}$

Step 20.10

Rewrite $- (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2}) + 3)$ as $- 1 (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2}) + 3)$ .

$\frac{\csc^{2} (\frac{x}{2}) (- 1 (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2}) + 3))}{{(3 x + 4)}^{2}}$

Step 20.11

Move the negative in front of the fraction.

$- \frac{\csc^{2} (\frac{x}{2}) (3 x \cot (\frac{x}{2}) + 4 \cot (\frac{x}{2}) + 3)}{{(3 x + 4)}^{2}}$