Calculus Examples

$r = \csc (θ) \cot (θ)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d θ} (r) = \frac{d}{d θ} (\csc (θ) \cot (θ))$

Step 2

The derivative of $r$ with respect to $θ$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (θ) = \cot (θ)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $2$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 3.11

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$r' = - \cot^{2} (θ) \csc (θ) - \csc^{3} (θ)$

Step 5

Replace $r'$ with $\frac{d r}{d θ}$ .

$\frac{d r}{d θ} = - \cot^{2} (θ) \csc (θ) - \csc^{3} (θ)$