Calculus Examples

$y = (\csc (x) + \cot (x)) (\csc (x) - \cot (x))$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \csc (x) + \cot (x)$ and $g (x) = \csc (x) - \cot (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate $\frac{d}{d x} [- \cot (x)]$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $- 1$ by $- 1$ .

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

Step 4.4

Multiply $\csc^{2} (x)$ by $1$ .

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

Step 5

Differentiate using the Sum Rule.

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Step 5.1

Reorder terms.

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

Step 5.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

Reorder terms.

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

Step 9

Simplify.

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Step 9.1

Reorder terms.

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

Step 9.2

Simplify each term.

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Step 9.2.1

Expand $(- \cot (x) \csc (x) + \csc^{2} (x)) (\csc (x) + \cot (x))$ using the FOIL Method.

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Step 9.2.1.1

Apply the distributive property.

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

Step 9.2.1.2

Apply the distributive property.

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

Step 9.2.1.3

Apply the distributive property.

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

Step 9.2.2

Simplify and combine like terms.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

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

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Step 9.2.2.1.1.1

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

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

Step 9.2.2.1.1.2

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

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

Step 9.2.2.1.1.3

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

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

Step 9.2.2.1.1.4

Add $1$ and $1$ .

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

Step 9.2.2.1.2

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

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Step 9.2.2.1.2.1

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

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

Step 9.2.2.1.2.2

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

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

Step 9.2.2.1.2.3

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

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

Step 9.2.2.1.2.4

Add $1$ and $1$ .

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

Step 9.2.2.1.3

Multiply $\csc^{2} (x)$ by $\csc (x)$ by adding the exponents.

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Step 9.2.2.1.3.1

Multiply $\csc^{2} (x)$ by $\csc (x)$ .

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Step 9.2.2.1.3.1.1

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

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

Step 9.2.2.1.3.1.2

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

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

Step 9.2.2.1.3.2

Add $2$ and $1$ .

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

Step 9.2.2.2

Reorder the factors of $- \cot (x) \csc^{2} (x)$ .

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

Step 9.2.2.3

Add $- \csc^{2} (x) \cot (x)$ and $\csc^{2} (x) \cot (x)$ .

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

Step 9.2.2.4

Subtract $\cot^{2} (x) \csc (x)$ from $0$ .

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

Step 9.2.3

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

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Step 9.2.3.1

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

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

Step 9.2.3.2

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

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

Step 9.2.3.3

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

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

Step 9.2.4

Reorder $- \cot^{2} (x)$ and $\csc^{2} (x)$ .

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

Step 9.2.5

Apply pythagorean identity.

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

Step 9.2.6

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

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

Step 9.2.7

Expand $(- \cot (x) \csc (x) - \csc^{2} (x)) (\csc (x) - \cot (x))$ using the FOIL Method.

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Step 9.2.7.1

Apply the distributive property.

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

Step 9.2.7.2

Apply the distributive property.

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

Step 9.2.7.3

Apply the distributive property.

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

Step 9.2.8

Simplify and combine like terms.

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Step 9.2.8.1

Simplify each term.

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Step 9.2.8.1.1

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

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Step 9.2.8.1.1.1

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

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

Step 9.2.8.1.1.2

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

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

Step 9.2.8.1.1.3

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

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

Step 9.2.8.1.1.4

Add $1$ and $1$ .

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

Step 9.2.8.1.2

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

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Step 9.2.8.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2.8.1.2.2

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

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

Step 9.2.8.1.2.3

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

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

Step 9.2.8.1.2.4

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

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

Step 9.2.8.1.2.5

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

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

Step 9.2.8.1.2.6

Add $1$ and $1$ .

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

Step 9.2.8.1.3

Multiply $\csc^{2} (x)$ by $\csc (x)$ by adding the exponents.

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Step 9.2.8.1.3.1

Move $\csc (x)$ .

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

Step 9.2.8.1.3.2

Multiply $\csc (x)$ by $\csc^{2} (x)$ .

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Step 9.2.8.1.3.2.1

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

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

Step 9.2.8.1.3.2.2

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

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

Step 9.2.8.1.3.3

Add $1$ and $2$ .

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

Step 9.2.8.1.4

Multiply $- \csc^{2} (x) (- \cot (x))$ .

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Step 9.2.8.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2.8.1.4.2

Multiply $\csc^{2} (x)$ by $1$ .

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

Step 9.2.8.2

Reorder the factors of $- \cot (x) \csc^{2} (x)$ .

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

Step 9.2.8.3

Add $- \csc^{2} (x) \cot (x)$ and $\csc^{2} (x) \cot (x)$ .

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

Step 9.2.8.4

Add $0$ and $\cot^{2} (x) \csc (x)$ .

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

Step 9.2.9

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

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Step 9.2.9.1

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

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

Step 9.2.9.2

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

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

Step 9.2.9.3

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

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

Step 9.2.10

Reorder $\cot^{2} (x)$ and $- \csc^{2} (x)$ .

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

Step 9.2.11

Factor $- 1$ out of $- \csc^{2} (x)$ .

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

Step 9.2.12

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

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

Step 9.2.13

Factor $- 1$ out of $- (\csc^{2} (x)) - 1 (- \cot^{2} (x))$ .

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

Step 9.2.14

Apply pythagorean identity.

$\csc (x) + \csc (x) (- 1 \cdot 1)$

Step 9.2.15

Multiply $- 1$ by $1$ .

$\csc (x) + \csc (x) \cdot - 1$

Step 9.2.16

Move $- 1$ to the left of $\csc (x)$ .

$\csc (x) - 1 \cdot \csc (x)$

Step 9.2.17

Rewrite $- 1 \csc (x)$ as $- \csc (x)$ .

$\csc (x) - \csc (x)$

Step 9.3

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

$0$