Calculus Examples

$y = \csc (θ) (θ + \cot (θ))$

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

$\csc (θ) (1 + \frac{d}{d θ} [\cot (θ)]) + (θ + \cot (θ)) \frac{d}{d θ} [\csc (θ)]$

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $\csc (θ)$ by $1$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

Add $1$ and $2$ .

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

Step 5.3.5

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

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

Step 5.3.6

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

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

Step 5.3.7

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

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

Step 5.3.8

Add $1$ and $1$ .

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

Step 5.4

Reorder terms.

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

Step 5.5

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

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Step 5.5.1

Factor $\csc (θ)$ out of $- θ \cot (θ) \csc (θ)$ .

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

Step 5.5.2

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

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

Step 5.5.3

Multiply by $1$ .

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

Step 5.5.4

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

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

Step 5.5.5

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

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

Step 5.5.6

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

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

Step 5.5.7

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

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

Step 5.6

Reorder $1$ and $- \csc^{2} (θ)$ .

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Apply pythagorean identity.

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

Step 5.11

Subtract $\cot^{2} (θ)$ from $- \cot^{2} (θ)$ .

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

Step 5.12

Apply the distributive property.

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

Step 5.13

Rewrite using the commutative property of multiplication.

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

Step 5.14

Rewrite using the commutative property of multiplication.

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

Step 5.15

Reorder factors in $- \csc (θ) θ \cot (θ) - 2 \csc (θ) \cot^{2} (θ)$ .

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