Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $- 1$ .

$- \sin (x) + 1 (\csc (x) \cot (x))$

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (x)}{\sin (x)} \csc (x) - \sin (x)$

Step 4.2.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} - \sin (x)$

Step 4.2.3

Multiply $\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Step 4.2.3.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{1}{\sin (x)}$ .

$\frac{\cos (x)}{\sin (x) \sin (x)} - \sin (x)$

Step 4.2.3.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{\cos (x)}{\sin^{1} (x) \sin (x)} - \sin (x)$

Step 4.2.3.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{\cos (x)}{\sin^{1} (x) \sin^{1} (x)} - \sin (x)$

Step 4.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\cos (x)}{{\sin (x)}^{1 + 1}} - \sin (x)$

Step 4.2.3.5

Add $1$ and $1$ .

$\frac{\cos (x)}{\sin^{2} (x)} - \sin (x)$

Step 4.3

Simplify each term.

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Step 4.3.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\frac{\cos (x)}{\sin (x) \sin (x)} - \sin (x)$

Step 4.3.2

Separate fractions.

$\frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} - \sin (x)$

Step 4.3.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 4.3.4

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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