Calculus Examples

$y = 9 + \cot (x) - 2 \csc (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $9 + \cot (x) - 2 \csc (x)$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- 2 \csc (x)]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- 2 \csc (x)]$

Step 1.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [- 2 \csc (x)]$ .

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \csc (x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\csc (x)]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $- 2$ .

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

Step 4

Simplify.

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Step 4.1

Subtract $\csc^{2} (x)$ from $0$ .

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

Step 4.2

Reorder terms.

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