Calculus Examples

$g (x) = \frac{8 x^{6} - 6 x^{8}}{x^{4}}$

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 $f (x) = 8 x^{6} - 6 x^{8}$ and $g (x) = x^{4}$ .

$\frac{x^{4} \frac{d}{d x} [8 x^{6} - 6 x^{8}] - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2

By the Sum Rule, the derivative of $8 x^{6} - 6 x^{8}$ with respect to $x$ is $\frac{d}{d x} [8 x^{6}] + \frac{d}{d x} [- 6 x^{8}]$ .

$\frac{x^{4} (\frac{d}{d x} [8 x^{6}] + \frac{d}{d x} [- 6 x^{8}]) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [8 x^{6}]$ .

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{6}$ with respect to $x$ is $8 \frac{d}{d x} [x^{6}]$ .

$\frac{x^{4} (8 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 6 x^{8}]) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{x^{4} (8 (6 x^{5}) + \frac{d}{d x} [- 6 x^{8}]) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 3.3

Multiply $6$ by $8$ .

$\frac{x^{4} (48 x^{5} + \frac{d}{d x} [- 6 x^{8}]) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 4

Evaluate $\frac{d}{d x} [- 6 x^{8}]$ .

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Step 4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{8}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{8}]$ .

$\frac{x^{4} (48 x^{5} - 6 \frac{d}{d x} [x^{8}]) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{x^{4} (48 x^{5} - 6 (8 x^{7})) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 4.3

Multiply $8$ by $- 6$ .

$\frac{x^{4} (48 x^{5} - 48 x^{7}) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

Reorder terms.

$\frac{x^{4} (- 48 x^{7} + 48 x^{5}) - (8 x^{6} - 6 x^{8}) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 5.2

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

$\frac{x^{4} (- 48 x^{7} + 48 x^{5}) - (8 x^{6} - 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{x^{4} (- 48 x^{7}) + x^{4} (48 x^{5}) - (8 x^{6} - 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{x^{4} (- 48 x^{7}) + x^{4} (48 x^{5}) + (- (8 x^{6}) - (- 6 x^{8})) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.3

Apply the distributive property.

$\frac{x^{4} (- 48 x^{7}) + x^{4} (48 x^{5}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 48 x^{4} x^{7} + x^{4} (48 x^{5}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.2

Multiply $x^{4}$ by $x^{7}$ by adding the exponents.

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Step 6.4.1.2.1

Move $x^{7}$ .

$\frac{- 48 (x^{7} x^{4}) + x^{4} (48 x^{5}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.2.2

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

$\frac{- 48 x^{7 + 4} + x^{4} (48 x^{5}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.2.3

Add $7$ and $4$ .

$\frac{- 48 x^{11} + x^{4} (48 x^{5}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.3

Rewrite using the commutative property of multiplication.

$\frac{- 48 x^{11} + 48 x^{4} x^{5} - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.4

Multiply $x^{4}$ by $x^{5}$ by adding the exponents.

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Step 6.4.1.4.1

Move $x^{5}$ .

$\frac{- 48 x^{11} + 48 (x^{5} x^{4}) - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.4.2

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

$\frac{- 48 x^{11} + 48 x^{5 + 4} - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.4.3

Add $5$ and $4$ .

$\frac{- 48 x^{11} + 48 x^{9} - (8 x^{6}) (4 x^{3}) - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.5

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 6.4.1.5.1

Move $x^{3}$ .

$\frac{- 48 x^{11} + 48 x^{9} - (8 (x^{3} x^{6})) \cdot 4 - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.5.2

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

$\frac{- 48 x^{11} + 48 x^{9} - (8 x^{3 + 6}) \cdot 4 - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.5.3

Add $3$ and $6$ .

$\frac{- 48 x^{11} + 48 x^{9} - (8 x^{9}) \cdot 4 - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.6

Multiply $8$ by $- 1$ .

$\frac{- 48 x^{11} + 48 x^{9} - 8 x^{9} \cdot 4 - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.7

Multiply $4$ by $- 8$ .

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} - (- 6 x^{8}) (4 x^{3})}{{(x^{4})}^{2}}$

Step 6.4.1.8

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

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Step 6.4.1.8.1

Move $x^{3}$ .

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} - (- 6 (x^{3} x^{8})) \cdot 4}{{(x^{4})}^{2}}$

Step 6.4.1.8.2

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

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} - (- 6 x^{3 + 8}) \cdot 4}{{(x^{4})}^{2}}$

Step 6.4.1.8.3

Add $3$ and $8$ .

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} - (- 6 x^{11}) \cdot 4}{{(x^{4})}^{2}}$

Step 6.4.1.9

Multiply $- 6$ by $- 1$ .

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} + 6 x^{11} \cdot 4}{{(x^{4})}^{2}}$

Step 6.4.1.10

Multiply $4$ by $6$ .

$\frac{- 48 x^{11} + 48 x^{9} - 32 x^{9} + 24 x^{11}}{{(x^{4})}^{2}}$

Step 6.4.2

Add $- 48 x^{11}$ and $24 x^{11}$ .

$\frac{- 24 x^{11} + 48 x^{9} - 32 x^{9}}{{(x^{4})}^{2}}$

Step 6.4.3

Subtract $32 x^{9}$ from $48 x^{9}$ .

$\frac{- 24 x^{11} + 16 x^{9}}{{(x^{4})}^{2}}$

Step 6.5

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 6.5.1

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

$\frac{- 24 x^{11} + 16 x^{9}}{x^{4 \cdot 2}}$

Step 6.5.2

Multiply $4$ by $2$ .

$\frac{- 24 x^{11} + 16 x^{9}}{x^{8}}$

Step 6.6

Factor $8 x^{9}$ out of $- 24 x^{11} + 16 x^{9}$ .

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Step 6.6.1

Factor $8 x^{9}$ out of $- 24 x^{11}$ .

$\frac{8 x^{9} (- 3 x^{2}) + 16 x^{9}}{x^{8}}$

Step 6.6.2

Factor $8 x^{9}$ out of $16 x^{9}$ .

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

Step 6.6.3

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

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

Step 6.7

Cancel the common factor of $x^{9}$ and $x^{8}$ .

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Step 6.7.1

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

$\frac{x^{8} (8 x (- 3 x^{2} + 2))}{x^{8}}$

Step 6.7.2

Cancel the common factors.

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Step 6.7.2.1

Multiply by $1$ .

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

Step 6.7.2.2

Cancel the common factor.

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

Step 6.7.2.3

Rewrite the expression.

$\frac{8 x (- 3 x^{2} + 2)}{1}$

Step 6.7.2.4

Divide $8 x (- 3 x^{2} + 2)$ by $1$ .

$8 x (- 3 x^{2} + 2)$

Step 6.8

Apply the distributive property.

$8 x (- 3 x^{2}) + 8 x \cdot 2$

Step 6.9

Rewrite using the commutative property of multiplication.

$8 \cdot - 3 x \cdot x^{2} + 8 x \cdot 2$

Step 6.10

Multiply $2$ by $8$ .

$8 \cdot - 3 x \cdot x^{2} + 16 x$

Step 6.11

Simplify each term.

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Step 6.11.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.11.1.1

Move $x^{2}$ .

$8 \cdot - 3 (x^{2} x) + 16 x$

Step 6.11.1.2

Multiply $x^{2}$ by $x$ .

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Step 6.11.1.2.1

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

$8 \cdot - 3 (x^{2} x^{1}) + 16 x$

Step 6.11.1.2.2

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

$8 \cdot - 3 x^{2 + 1} + 16 x$

Step 6.11.1.3

Add $2$ and $1$ .

$8 \cdot - 3 x^{3} + 16 x$

Step 6.11.2

Multiply $8$ by $- 3$ .

$- 24 x^{3} + 16 x$