Calculus Examples

$\csc (x) \tan (x) \cos (x)$

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 $f (x) = \csc (x) \tan (x)$ and $g (x) = \cos (x)$ .

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

Step 2

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

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

Step 3

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 $f (x) = \csc (x)$ and $g (x) = \tan (x)$ .

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

Step 4

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) (\csc (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\csc (x)])$

Step 5

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

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) (\csc (x) \sec^{2} (x) + \tan (x) (- \csc (x) \cot (x)))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) (\csc (x) \sec^{2} (x)) + \cos (x) (\tan (x) (- \csc (x) \cot (x)))$

Step 6.2

Remove unnecessary parentheses.

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) \csc (x) \sec^{2} (x) + \cos (x) \tan (x) (- \csc (x) \cot (x))$

Step 6.3

Reorder terms.

$- \csc (x) \sin (x) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4

Simplify each term.

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Step 6.4.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 6.4.1.1

Add parentheses.

$- (\csc (x) \sin (x)) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.1.2

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

$- (\frac{1}{\sin (x)} \sin (x)) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.1.3

Cancel the common factors.

$- 1 \cdot 1 \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.2

Multiply $- 1$ by $1$ .

$- \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.3

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

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

Step 6.4.4

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

$- \frac{\sin (x)}{\cos (x)} + {(\frac{1}{\cos (x)})}^{2} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.5

Apply the product rule to $\frac{1}{\cos (x)}$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos^{2} (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.6

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

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Step 6.4.6.1

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.6.2

Cancel the common factor.

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.6.3

Rewrite the expression.

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x)} \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Step 6.4.7

One to any power is one.

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

Step 6.4.8

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

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

Step 6.4.9

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

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

Step 6.4.10

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

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

Step 6.4.11

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

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Step 6.4.11.1

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

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

Step 6.4.11.2

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

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

Step 6.4.11.3

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

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

Step 6.4.11.4

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

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

Step 6.4.11.5

Add $1$ and $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x)} \csc (x) \tan (x)$

Step 6.4.12

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \tan (x)$

Step 6.4.13

Multiply $- \frac{\cos^{2} (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Step 6.4.13.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{\cos^{2} (x)}{\sin (x)}$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x) \sin (x)} \tan (x)$

Step 6.4.13.2

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{1} (x) \sin (x)} \tan (x)$

Step 6.4.13.3

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{1} (x) \sin^{1} (x)} \tan (x)$

Step 6.4.13.4

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{{\sin (x)}^{1 + 1}} \tan (x)$

Step 6.4.13.5

Add $1$ and $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} \tan (x)$

Step 6.4.14

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 6.4.15

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

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Step 6.4.15.1

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{- \cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 6.4.15.2

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

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{\cos (x) (- \cos (x))}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 6.4.15.3

Cancel the common factor.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{\cos (x) (- \cos (x))}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 6.4.15.4

Rewrite the expression.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{- \cos (x)}{\sin^{2} (x)} \sin (x)$

Step 6.4.16

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

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Step 6.4.16.1

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

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

Step 6.4.16.2

Cancel the common factor.

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

Step 6.4.16.3

Rewrite the expression.

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

Step 6.4.17

Move the negative in front of the fraction.

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

Step 6.5

Simplify each term.

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Step 6.5.1

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

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

Step 6.5.2

Separate fractions.

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

Step 6.5.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 6.5.4

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

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

Step 6.5.5

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

$- \tan (x) + \csc (x) \sec (x) - \cot (x)$