Calculus Examples

f (x) = sin (x) csc (x)

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

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = sin (x)

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

g (x) = csc (x)

$g (x) = \csc (x)$ .

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

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

Step 2

The derivative of

csc (x)

$\csc (x)$ with respect to

x

$x$ is

- csc (x) cot (x)

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

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

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

Step 3

The derivative of

sin (x)

$\sin (x)$ with respect to

x

$x$ is

cos (x)

$\cos (x)$ .

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite

cot (x)

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

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

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

Step 4.2.2

Rewrite

csc (x)

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

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

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

Step 4.2.3

Multiply

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

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

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Step 4.2.3.1

Multiply

\frac{1}{sin (x)}

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

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

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

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

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

Step 4.2.3.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 4.2.3.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

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

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

$0$