Calculus Examples

$f (x) = \frac{1}{\csc^{5} (x)}$

Step 1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1

Rewrite $\frac{1}{\csc^{5} (x)}$ as ${(\csc^{5} (x))}^{- 1}$ .

$\frac{d}{d x} [{(\csc^{5} (x))}^{- 1}]$

Step 1.2

Multiply the exponents in ${(\csc^{5} (x))}^{- 1}$ .

Tap for more steps...

Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [{\csc (x)}^{5 \cdot - 1}]$

Step 1.2.2

Multiply $5$ by $- 1$ .

$\frac{d}{d x} [\csc^{- 5} (x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 5}$ and $g (x) = \csc (x)$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $\csc (x)$ .

$\frac{d}{d u} [u^{- 5}] \frac{d}{d x} [\csc (x)]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 5$ .

$- 5 u^{- 6} \frac{d}{d x} [\csc (x)]$

Step 2.3

Replace all occurrences of $u$ with $\csc (x)$ .

$- 5 \csc^{- 6} (x) \frac{d}{d x} [\csc (x)]$

Step 3

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

$- 5 \csc^{- 6} (x) (- \csc (x) \cot (x))$

Step 4

Multiply $- 1$ by $- 5$ .

$5 \csc^{- 6} (x) (\csc (x) \cot (x))$

Step 5

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

$5 (\csc^{1} (x) \csc^{- 6} (x)) \cot (x)$

Step 6

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

$5 {\csc (x)}^{1 - 6} \cot (x)$

Step 7

Subtract $6$ from $1$ .

$5 \csc^{- 5} (x) \cot (x)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 \frac{1}{\csc^{5} (x)} \cot (x)$

Step 8.2

Combine terms.

Tap for more steps...

Step 8.2.1

Combine $5$ and $\frac{1}{\csc^{5} (x)}$ .

$\frac{5}{\csc^{5} (x)} \cot (x)$

Step 8.2.2

Combine $\frac{5}{\csc^{5} (x)}$ and $\cot (x)$ .

$\frac{5 \cot (x)}{\csc^{5} (x)}$

Step 8.3

Factor $\csc (x)$ out of $\csc^{5} (x)$ .

$\frac{5 \cot (x)}{\csc (x) \csc^{4} (x)}$

Step 8.4

Separate fractions.

$\frac{5}{\csc^{4} (x)} \cdot \frac{\cot (x)}{\csc (x)}$

Step 8.5

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

$\frac{5}{\csc^{4} (x)} \cdot \frac{\cot (x)}{\frac{1}{\sin (x)}}$

Step 8.6

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

$\frac{5}{\csc^{4} (x)} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)}}$

Step 8.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{5}{\csc^{4} (x)} (\frac{\cos (x)}{\sin (x)} \sin (x))$

Step 8.8

Write $\sin (x)$ as a fraction with denominator $1$ .

$\frac{5}{\csc^{4} (x)} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1})$

Step 8.9

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

Tap for more steps...

Step 8.9.1

Cancel the common factor.

$\frac{5}{\csc^{4} (x)} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1})$

Step 8.9.2

Rewrite the expression.

$\frac{5}{\csc^{4} (x)} \cos (x)$

Step 8.10

Combine $\frac{5}{\csc^{4} (x)}$ and $\cos (x)$ .

$\frac{5 \cos (x)}{\csc^{4} (x)}$

Step 8.11

Factor $\csc (x)$ out of $\csc^{4} (x)$ .

$\frac{5 \cos (x)}{\csc (x) \csc^{3} (x)}$

Step 8.12

Separate fractions.

$\frac{5}{\csc^{3} (x)} \cdot \frac{\cos (x)}{\csc (x)}$

Step 8.13

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

$\frac{5}{\csc^{3} (x)} \cdot \frac{\cos (x)}{\frac{1}{\sin (x)}}$

Step 8.14

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{5}{\csc^{3} (x)} (\cos (x) \sin (x))$

Step 8.15

Multiply $\frac{5}{\csc^{3} (x)} (\cos (x) \sin (x))$ .

Tap for more steps...

Step 8.15.1

Combine $\cos (x)$ and $\frac{5}{\csc^{3} (x)}$ .

$\frac{\cos (x) \cdot 5}{\csc^{3} (x)} \sin (x)$

Step 8.15.2

Combine $\frac{\cos (x) \cdot 5}{\csc^{3} (x)}$ and $\sin (x)$ .

$\frac{\cos (x) \cdot 5 \sin (x)}{\csc^{3} (x)}$

Step 8.16

Factor $\csc (x)$ out of $\csc^{3} (x)$ .

$\frac{\cos (x) \cdot 5 \sin (x)}{\csc (x) \csc^{2} (x)}$

Step 8.17

Separate fractions.

$\frac{5 \sin (x)}{\csc^{2} (x)} \cdot \frac{\cos (x)}{\csc (x)}$

Step 8.18

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

$\frac{5 \sin (x)}{\csc^{2} (x)} \cdot \frac{\cos (x)}{\frac{1}{\sin (x)}}$

Step 8.19

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{5 \sin (x)}{\csc^{2} (x)} (\cos (x) \sin (x))$

Step 8.20

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

$\frac{5 \sin (x)}{\csc (x) \csc (x)} (\cos (x) \sin (x))$

Step 8.21

Separate fractions.

$\frac{5}{\csc (x)} \cdot \frac{\sin (x)}{\csc (x)} (\cos (x) \sin (x))$

Step 8.22

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

$\frac{5}{\csc (x)} \cdot \frac{\sin (x)}{\frac{1}{\sin (x)}} (\cos (x) \sin (x))$

Step 8.23

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{5}{\csc (x)} (\sin (x) \sin (x)) (\cos (x) \sin (x))$

Step 8.24

Simplify.

Tap for more steps...

Step 8.24.1

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

$\frac{5}{\csc (x)} (\sin^{1} (x) \sin (x)) (\cos (x) \sin (x))$

Step 8.24.2

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

$\frac{5}{\csc (x)} (\sin^{1} (x) \sin^{1} (x)) (\cos (x) \sin (x))$

Step 8.24.3

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

$\frac{5}{\csc (x)} {\sin (x)}^{1 + 1} (\cos (x) \sin (x))$

Step 8.24.4

Add $1$ and $1$ .

$\frac{5}{\csc (x)} \sin^{2} (x) (\cos (x) \sin (x))$

Step 8.25

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

Tap for more steps...

Step 8.25.1

Move $\sin (x)$ .

$\frac{5}{\csc (x)} (\sin (x) \sin^{2} (x)) \cos (x)$

Step 8.25.2

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

Tap for more steps...

Step 8.25.2.1

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

$\frac{5}{\csc (x)} (\sin^{1} (x) \sin^{2} (x)) \cos (x)$

Step 8.25.2.2

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

$\frac{5}{\csc (x)} {\sin (x)}^{1 + 2} \cos (x)$

Step 8.25.3

Add $1$ and $2$ .

$\frac{5}{\csc (x)} \sin^{3} (x) \cos (x)$

Step 8.26

Separate fractions.

$\frac{5}{1} \cdot \frac{1}{\csc (x)} \sin^{3} (x) \cos (x)$

Step 8.27

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

$\frac{5}{1} \cdot \frac{1}{\frac{1}{\sin (x)}} \sin^{3} (x) \cos (x)$

Step 8.28

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{5}{1} (1 \sin (x)) \sin^{3} (x) \cos (x)$

Step 8.29

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

$\frac{5}{1} \sin (x) \sin^{3} (x) \cos (x)$

Step 8.30

Multiply $\sin (x)$ by $\sin^{3} (x)$ by adding the exponents.

Tap for more steps...

Step 8.30.1

Move $\sin^{3} (x)$ .

$\frac{5}{1} (\sin^{3} (x) \sin (x)) \cos (x)$

Step 8.30.2

Multiply $\sin^{3} (x)$ by $\sin (x)$ .

Tap for more steps...

Step 8.30.2.1

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

$\frac{5}{1} (\sin^{3} (x) \sin^{1} (x)) \cos (x)$

Step 8.30.2.2

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

$\frac{5}{1} {\sin (x)}^{3 + 1} \cos (x)$

Step 8.30.3

Add $3$ and $1$ .

$\frac{5}{1} \sin^{4} (x) \cos (x)$

Step 8.31

Divide $5$ by $1$ .

$5 \sin^{4} (x) \cos (x)$