Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

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

Step 3.10.2

Simplify the numerator.

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Step 3.10.2.1

Simplify each term.

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Step 3.10.2.1.1

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

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

Step 3.10.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.10.2.1.3

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

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Step 3.10.2.1.3.1

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

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

Step 3.10.2.1.3.2

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

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

Step 3.10.2.1.3.3

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

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

Step 3.10.2.1.3.4

Add $1$ and $1$ .

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

Step 3.10.2.2

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

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Step 3.10.2.2.1

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

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

Step 3.10.2.2.2

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

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

Step 3.10.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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