Calculus Examples

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

Step 1

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

$\frac{d}{d x} [- \csc (x)] + \frac{d}{d x} [- \sin (x)]$

Step 2

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

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Step 2.1

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

$- \frac{d}{d x} [\csc (x)] + \frac{d}{d x} [- \sin (x)]$

Step 2.2

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

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

Step 2.3

Multiply $- 1$ by $- 1$ .

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

Step 2.4

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

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

Step 3

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

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Step 3.1

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

$\csc (x) \cot (x) - \frac{d}{d x} [\sin (x)]$

Step 3.2

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

$\csc (x) \cot (x) - \cos (x)$