Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

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

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Simplify the numerator.

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Step 3.6.2.1

Combine the opposite terms in $1 (- \csc^{2} (x)) + \cot (x) (- \csc^{2} (x)) + \cot (x) \csc^{2} (x)$ .

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Step 3.6.2.1.1

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

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

Step 3.6.2.1.2

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

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

Step 3.6.2.1.3

Add $1 (- \csc^{2} (x))$ and $0$ .

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

Step 3.6.2.2

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

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

Step 3.6.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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