Calculus Examples

$y = 5 \csc (x) + 5 \cos (x)$

Step 1

By the Sum Rule, the derivative of $5 \csc (x) + 5 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [5 \csc (x)] + \frac{d}{d x} [5 \cos (x)]$ .

$\frac{d}{d x} [5 \csc (x)] + \frac{d}{d x} [5 \cos (x)]$

Step 2

Evaluate $\frac{d}{d x} [5 \csc (x)]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \csc (x)$ with respect to $x$ is $5 \frac{d}{d x} [\csc (x)]$ .

$5 \frac{d}{d x} [\csc (x)] + \frac{d}{d x} [5 \cos (x)]$

Step 2.2

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

$5 (- \csc (x) \cot (x)) + \frac{d}{d x} [5 \cos (x)]$

Step 2.3

Multiply $- 1$ by $5$ .

$- 5 \csc (x) \cot (x) + \frac{d}{d x} [5 \cos (x)]$

Step 3

Evaluate $\frac{d}{d x} [5 \cos (x)]$ .

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos (x)]$ .

$- 5 \csc (x) \cot (x) + 5 \frac{d}{d x} [\cos (x)]$

Step 3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- 5 \csc (x) \cot (x) + 5 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $5$ .

$- 5 \csc (x) \cot (x) - 5 \sin (x)$