Calculus Examples

$\frac{8 x}{1 - \cot (x)}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x}{1 - \cot (x)}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x}{1 - \cot (x)}]$ .

$8 \frac{d}{d x} [\frac{x}{1 - \cot (x)}]$

Step 2

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

$8 \frac{(1 - \cot (x)) \frac{d}{d x} [x] - x \frac{d}{d x} [1 - \cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$8 \frac{(1 - \cot (x)) \cdot 1 - x \frac{d}{d x} [1 - \cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 3.2

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

$8 \frac{1 - \cot (x) - x \frac{d}{d x} [1 - \cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 3.3

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

$8 \frac{1 - \cot (x) - x (\frac{d}{d x} [1] + \frac{d}{d x} [- \cot (x)])}{{(1 - \cot (x))}^{2}}$

Step 3.4

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

$8 \frac{1 - \cot (x) - x (0 + \frac{d}{d x} [- \cot (x)])}{{(1 - \cot (x))}^{2}}$

Step 3.5

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

$8 \frac{1 - \cot (x) - x \frac{d}{d x} [- \cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cot (x)$ with respect to $x$ is $- \frac{d}{d x} [\cot (x)]$ .

$8 \frac{1 - \cot (x) - x (- \frac{d}{d x} [\cot (x)])}{{(1 - \cot (x))}^{2}}$

Step 3.7

Multiply.

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

$8 \frac{1 - \cot (x) + 1 x \frac{d}{d x} [\cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 3.7.2

Multiply $x$ by $1$ .

$8 \frac{1 - \cot (x) + x \frac{d}{d x} [\cot (x)]}{{(1 - \cot (x))}^{2}}$

Step 4

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

$8 \frac{1 - \cot (x) + x (- \csc^{2} (x))}{{(1 - \cot (x))}^{2}}$

Step 5

Combine $8$ and $\frac{1 - \cot (x) + x (- \csc^{2} (x))}{{(1 - \cot (x))}^{2}}$ .

$\frac{8 (1 - \cot (x) + x (- \csc^{2} (x)))}{{(1 - \cot (x))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{8 \cdot 1 + 8 (- \cot (x)) + 8 (x (- \csc^{2} (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $8$ by $1$ .

$\frac{8 + 8 (- \cot (x)) + 8 (x (- \csc^{2} (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.2.2

Multiply $- 1$ by $8$ .

$\frac{8 - 8 \cot (x) + 8 (x (- \csc^{2} (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.2.3

Rewrite using the commutative property of multiplication.

$\frac{8 - 8 \cot (x) + 8 (- x \csc^{2} (x))}{{(1 - \cot (x))}^{2}}$

Step 6.2.4

Multiply $- 1$ by $8$ .

$\frac{8 - 8 \cot (x) - 8 x \csc^{2} (x)}{{(1 - \cot (x))}^{2}}$

Step 6.3

Reorder terms.

$\frac{- 8 x \csc^{2} (x) + 8 - 8 \cot (x)}{{(1 - \cot (x))}^{2}}$

Step 6.4

Factor $8$ out of $- 8 x \csc^{2} (x) + 8 - 8 \cot (x)$ .

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Step 6.4.1

Factor $8$ out of $- 8 x \csc^{2} (x)$ .

$\frac{8 (- x \csc^{2} (x)) + 8 - 8 \cot (x)}{{(1 - \cot (x))}^{2}}$

Step 6.4.2

Factor $8$ out of $8$ .

$\frac{8 (- x \csc^{2} (x)) + 8 (1) - 8 \cot (x)}{{(1 - \cot (x))}^{2}}$

Step 6.4.3

Factor $8$ out of $- 8 \cot (x)$ .

$\frac{8 (- x \csc^{2} (x)) + 8 (1) + 8 (- \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.4.4

Factor $8$ out of $8 (- x \csc^{2} (x)) + 8 (1)$ .

$\frac{8 (- x \csc^{2} (x) + 1) + 8 (- \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.4.5

Factor $8$ out of $8 (- x \csc^{2} (x) + 1) + 8 (- \cot (x))$ .

$\frac{8 (- x \csc^{2} (x) + 1 - \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.5

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

$\frac{8 (- (x \csc^{2} (x)) + 1 - \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.6

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

$\frac{8 (- (x \csc^{2} (x)) - 1 (- 1) - \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.7

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

$\frac{8 (- (x \csc^{2} (x) - 1) - \cot (x))}{{(1 - \cot (x))}^{2}}$

Step 6.8

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

$\frac{8 (- (x \csc^{2} (x) - 1) - (\cot (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.9

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

$\frac{8 (- (x \csc^{2} (x) - 1 + \cot (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.10

Rewrite $- (x \csc^{2} (x) - 1 + \cot (x))$ as $- 1 (x \csc^{2} (x) - 1 + \cot (x))$ .

$\frac{8 (- 1 (x \csc^{2} (x) - 1 + \cot (x)))}{{(1 - \cot (x))}^{2}}$

Step 6.11

Move the negative in front of the fraction.

$- \frac{8 (x \csc^{2} (x) - 1 + \cot (x))}{{(1 - \cot (x))}^{2}}$