Calculus Examples

$y = \frac{1 + \csc (x)}{1 - \csc (x)}$

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 $f (x) = 1 + \csc (x)$ and $g (x) = 1 - \csc (x)$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [\csc (x)]$ .

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

Step 3

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

$\frac{(1 - \csc (x)) (- \csc (x) \cot (x)) - (1 + \csc (x)) (0 + \frac{d}{d x} [- \csc (x)])}{{(1 - \csc (x))}^{2}}$

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Multiply.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.2

Multiply $1 + \csc (x)$ by $1$ .

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

Step 5

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $- \csc (x) \cot (x)$ by $1$ .

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

Step 6.3.1.2

Multiply $- \csc (x) (- \csc (x) \cot (x))$ .

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Step 6.3.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.3.1.2.2

Multiply $\csc (x)$ by $1$ .

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

Step 6.3.1.2.3

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

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

Step 6.3.1.2.4

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

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

Step 6.3.1.2.5

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

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

Step 6.3.1.2.6

Add $1$ and $1$ .

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

Step 6.3.1.3

Multiply $- \csc (x) \cot (x)$ by $1$ .

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

Step 6.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.5

Multiply $- \csc (x) \csc (x)$ .

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Step 6.3.1.5.1

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

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

Step 6.3.1.5.2

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

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

Step 6.3.1.5.3

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

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

Step 6.3.1.5.4

Add $1$ and $1$ .

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

Step 6.3.2

Combine the opposite terms in $- \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - \csc (x) \cot (x) - \csc^{2} (x) \cot (x)$ .

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Step 6.3.2.1

Subtract $\csc^{2} (x) \cot (x)$ from $\csc^{2} (x) \cot (x)$ .

$\frac{- \csc (x) \cot (x) - \csc (x) \cot (x) + 0}{{(1 - \csc (x))}^{2}}$

Step 6.3.2.2

Add $- \csc (x) \cot (x) - \csc (x) \cot (x)$ and $0$ .

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

Step 6.3.3

Subtract $\csc (x) \cot (x)$ from $- \csc (x) \cot (x)$ .

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

Step 6.4

Move the negative in front of the fraction.

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