Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply.

Tap for more steps...

Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 3.10

Simplify.

Tap for more steps...

Step 3.10.1

Apply the distributive property.

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

Step 3.10.2

Simplify the numerator.

Tap for more steps...

Step 3.10.2.1

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

Tap for more steps...

Step 3.10.2.1.1

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

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

Step 3.10.2.1.2

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

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

Step 3.10.2.1.3

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

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

Step 3.10.2.1.4

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

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

Step 3.10.2.2

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

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

Step 3.10.2.3

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

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

Step 3.10.2.4

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

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

Step 3.10.2.5

Apply pythagorean identity.

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

Step 3.10.2.6

Simplify each term.

Tap for more steps...

Step 3.10.2.6.1

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

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

Step 3.10.2.6.2

Multiply $- 1$ by $1$ .

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

Step 3.10.2.7

Apply the distributive property.

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

Step 3.10.2.8

Rewrite using the commutative property of multiplication.

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

Step 3.10.2.9

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

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

Step 3.10.2.10

Simplify each term.

Tap for more steps...

Step 3.10.2.10.1

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

Tap for more steps...

Step 3.10.2.10.1.1

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

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

Step 3.10.2.10.1.2

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

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

Step 3.10.2.10.1.3

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

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

Step 3.10.2.10.1.4

Add $1$ and $1$ .

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

Step 3.10.2.10.2

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

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

Step 3.10.3

Simplify the numerator.

Tap for more steps...

Step 3.10.3.1

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

Tap for more steps...

Step 3.10.3.1.1

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

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

Step 3.10.3.1.2

Factor $\csc (x)$ out of $- \csc (x)$ .

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

Step 3.10.3.1.3

Factor $\csc (x)$ out of $\csc (x) (- \csc (x)) + \csc (x) \cdot - 1$ .

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

Step 3.10.3.2

Factor $- 1$ out of $- \csc (x) - 1$ .

Tap for more steps...

Step 3.10.3.2.1

Factor $- 1$ out of $- \csc (x)$ .

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

Step 3.10.3.2.2

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

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

Step 3.10.3.2.3

Factor $- 1$ out of $- (\csc (x)) - 1 (1)$ .

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

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

Step 3.10.3.3

Factor out negative.

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

Step 3.10.4

Cancel the common factor of $\csc (x) + 1$ and ${(1 + \csc (x))}^{2}$ .

Tap for more steps...

Step 3.10.4.1

Reorder terms.

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

Step 3.10.4.2

Factor $1 + \csc (x)$ out of $- \csc (x) (1 + \csc (x))$ .

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

Step 3.10.4.3

Cancel the common factors.

Tap for more steps...

Step 3.10.4.3.1

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

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

Step 3.10.4.3.2

Cancel the common factor.

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

Step 3.10.4.3.3

Rewrite the expression.

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

Step 3.10.5

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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