Calculus Examples

$\frac{4 - 2 x^{4} + x^{5}}{x^{9}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 - 2 x^{4} + x^{5}$ and $g (x) = x^{9}$ .

$\frac{x^{9} \frac{d}{d x} [4 - 2 x^{4} + x^{5}] - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{{(x^{9})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(x^{9})}^{2}$ .

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Step 2.1.1

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

$\frac{x^{9} \frac{d}{d x} [4 - 2 x^{4} + x^{5}] - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{9 \cdot 2}}$

Step 2.1.2

Multiply $9$ by $2$ .

$\frac{x^{9} \frac{d}{d x} [4 - 2 x^{4} + x^{5}] - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.2

By the Sum Rule, the derivative of $4 - 2 x^{4} + x^{5}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [x^{5}]$ .

$\frac{x^{9} (\frac{d}{d x} [4] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x^{9} (0 + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.4

Add $0$ and $\frac{d}{d x} [- 2 x^{4}]$ .

$\frac{x^{9} (\frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4}]$ .

$\frac{x^{9} (- 2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{9} (- 2 (4 x^{3}) + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.7

Multiply $4$ by $- 2$ .

$\frac{x^{9} (- 8 x^{3} + \frac{d}{d x} [x^{5}]) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.8

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

$\frac{x^{9} (- 8 x^{3} + 5 x^{4}) - (4 - 2 x^{4} + x^{5}) \frac{d}{d x} [x^{9}]}{x^{18}}$

Step 2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$\frac{x^{9} (- 8 x^{3} + 5 x^{4}) - (4 - 2 x^{4} + x^{5}) (9 x^{8})}{x^{18}}$

Step 2.10

Simplify with factoring out.

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Step 2.10.1

Multiply $9$ by $- 1$ .

$\frac{x^{9} (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5}) x^{8}}{x^{18}}$

Step 2.10.2

Factor $x^{8}$ out of $x^{9} (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5}) x^{8}$ .

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Step 2.10.2.1

Factor $x^{8}$ out of $x^{9} (- 8 x^{3} + 5 x^{4})$ .

$\frac{x^{8} (x (- 8 x^{3} + 5 x^{4})) - 9 (4 - 2 x^{4} + x^{5}) x^{8}}{x^{18}}$

Step 2.10.2.2

Factor $x^{8}$ out of $- 9 (4 - 2 x^{4} + x^{5}) x^{8}$ .

$\frac{x^{8} (x (- 8 x^{3} + 5 x^{4})) + x^{8} (- 9 (4 - 2 x^{4} + x^{5}))}{x^{18}}$

Step 2.10.2.3

Factor $x^{8}$ out of $x^{8} (x (- 8 x^{3} + 5 x^{4})) + x^{8} (- 9 (4 - 2 x^{4} + x^{5}))$ .

$\frac{x^{8} (x (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5}))}{x^{18}}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $x^{8}$ out of $x^{18}$ .

$\frac{x^{8} (x (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5}))}{x^{8} x^{10}}$

Step 3.2

Cancel the common factor.

$\frac{x^{8} (x (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5}))}{x^{8} x^{10}}$

Step 3.3

Rewrite the expression.

$\frac{x (- 8 x^{3} + 5 x^{4}) - 9 (4 - 2 x^{4} + x^{5})}{x^{10}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x (- 8 x^{3}) + x (5 x^{4}) - 9 (4 - 2 x^{4} + x^{5})}{x^{10}}$

Step 4.2

Apply the distributive property.

$\frac{x (- 8 x^{3}) + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 8 x \cdot x^{3} + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.3.1.2.1

Move $x^{3}$ .

$\frac{- 8 (x^{3} x) + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.2.2

Multiply $x^{3}$ by $x$ .

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Step 4.3.1.2.2.1

Raise $x$ to the power of $1$ .

$\frac{- 8 (x^{3} x^{1}) + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.2.2.2

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

$\frac{- 8 x^{3 + 1} + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.2.3

Add $3$ and $1$ .

$\frac{- 8 x^{4} + x (5 x^{4}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{- 8 x^{4} + 5 x \cdot x^{4} - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.4

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 4.3.1.4.1

Move $x^{4}$ .

$\frac{- 8 x^{4} + 5 (x^{4} x) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.4.2

Multiply $x^{4}$ by $x$ .

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Step 4.3.1.4.2.1

Raise $x$ to the power of $1$ .

$\frac{- 8 x^{4} + 5 (x^{4} x^{1}) - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.4.2.2

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

$\frac{- 8 x^{4} + 5 x^{4 + 1} - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.4.3

Add $4$ and $1$ .

$\frac{- 8 x^{4} + 5 x^{5} - 9 \cdot 4 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.5

Multiply $- 9$ by $4$ .

$\frac{- 8 x^{4} + 5 x^{5} - 36 - 9 (- 2 x^{4}) - 9 x^{5}}{x^{10}}$

Step 4.3.1.6

Multiply $- 2$ by $- 9$ .

$\frac{- 8 x^{4} + 5 x^{5} - 36 + 18 x^{4} - 9 x^{5}}{x^{10}}$

Step 4.3.2

Add $- 8 x^{4}$ and $18 x^{4}$ .

$\frac{10 x^{4} + 5 x^{5} - 36 - 9 x^{5}}{x^{10}}$

Step 4.3.3

Subtract $9 x^{5}$ from $5 x^{5}$ .

$\frac{10 x^{4} - 4 x^{5} - 36}{x^{10}}$

Step 4.4

Reorder terms.

$\frac{- 4 x^{5} + 10 x^{4} - 36}{x^{10}}$

Step 4.5

Factor $2$ out of $- 4 x^{5} + 10 x^{4} - 36$ .

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Step 4.5.1

Factor $2$ out of $- 4 x^{5}$ .

$\frac{2 (- 2 x^{5}) + 10 x^{4} - 36}{x^{10}}$

Step 4.5.2

Factor $2$ out of $10 x^{4}$ .

$\frac{2 (- 2 x^{5}) + 2 (5 x^{4}) - 36}{x^{10}}$

Step 4.5.3

Factor $2$ out of $- 36$ .

$\frac{2 (- 2 x^{5}) + 2 (5 x^{4}) + 2 (- 18)}{x^{10}}$

Step 4.5.4

Factor $2$ out of $2 (- 2 x^{5}) + 2 (5 x^{4})$ .

$\frac{2 (- 2 x^{5} + 5 x^{4}) + 2 (- 18)}{x^{10}}$

Step 4.5.5

Factor $2$ out of $2 (- 2 x^{5} + 5 x^{4}) + 2 (- 18)$ .

$\frac{2 (- 2 x^{5} + 5 x^{4} - 18)}{x^{10}}$

Step 4.6

Factor $- 1$ out of $- 2 x^{5}$ .

$\frac{2 (- (2 x^{5}) + 5 x^{4} - 18)}{x^{10}}$

Step 4.7

Factor $- 1$ out of $5 x^{4}$ .

$\frac{2 (- (2 x^{5}) - (- 5 x^{4}) - 18)}{x^{10}}$

Step 4.8

Factor $- 1$ out of $- (2 x^{5}) - (- 5 x^{4})$ .

$\frac{2 (- (2 x^{5} - 5 x^{4}) - 18)}{x^{10}}$

Step 4.9

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{2 (- (2 x^{5} - 5 x^{4}) - 1 (18))}{x^{10}}$

Step 4.10

Factor $- 1$ out of $- (2 x^{5} - 5 x^{4}) - 1 (18)$ .

$\frac{2 (- (2 x^{5} - 5 x^{4} + 18))}{x^{10}}$

Step 4.11

Rewrite $- (2 x^{5} - 5 x^{4} + 18)$ as $- 1 (2 x^{5} - 5 x^{4} + 18)$ .

$\frac{2 (- 1 (2 x^{5} - 5 x^{4} + 18))}{x^{10}}$

Step 4.12

Move the negative in front of the fraction.

$- \frac{2 (2 x^{5} - 5 x^{4} + 18)}{x^{10}}$