Calculus Examples

$f (x) = \sin (x) \csc (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \csc (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

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

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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

Step 4.2.3.4

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

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

Step 4.2.3.5

Add $1$ and $1$ .

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

Step 4.2.4

Cancel the common factor of $\sin (x)$ .

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Step 4.2.4.1

Move the leading negative in $- \frac{\cos (x)}{\sin^{2} (x)}$ into the numerator.

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

Step 4.2.4.2

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

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

Step 4.2.4.3

Cancel the common factor.

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

Step 4.2.4.4

Rewrite the expression.

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

Step 4.2.5

Move the negative in front of the fraction.

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

Step 4.2.6

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

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

Step 4.2.7

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 4.3

Add $- \frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

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