Algebra Examples

$f (x) = \frac{2^{- 8} x^{- 4} y^{2}}{2^{- 5} x^{- 7} y^{4}}$

Step 1

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

$\frac{d}{d x} [\frac{\frac{1}{2^{8}} x^{- 4} y^{2}}{2^{- 5} x^{- 7} y^{4}}]$

Step 2

Raise $2$ to the power of $8$ .

$\frac{d}{d x} [\frac{\frac{1}{256} x^{- 4} y^{2}}{2^{- 5} x^{- 7} y^{4}}]$

Step 3

Combine $\frac{1}{256}$ and $x^{- 4}$ .

$\frac{d}{d x} [\frac{\frac{x^{- 4}}{256} y^{2}}{2^{- 5} x^{- 7} y^{4}}]$

Step 4

Combine $\frac{x^{- 4}}{256}$ and $y^{2}$ .

$\frac{d}{d x} [\frac{\frac{x^{- 4} y^{2}}{256}}{2^{- 5} x^{- 7} y^{4}}]$

Step 5

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{2^{- 5} x^{- 7} y^{4}}]$

Step 6

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

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{\frac{1}{2^{5}} x^{- 7} y^{4}}]$

Step 7

Raise $2$ to the power of $5$ .

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{\frac{1}{32} x^{- 7} y^{4}}]$

Step 8

Combine $\frac{1}{32}$ and $x^{- 7}$ .

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{\frac{x^{- 7}}{32} y^{4}}]$

Step 9

Combine $\frac{x^{- 7}}{32}$ and $y^{4}$ .

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{\frac{x^{- 7} y^{4}}{32}}]$

Step 10

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [\frac{\frac{y^{2}}{256 x^{4}}}{\frac{y^{4}}{32 x^{7}}}]$

Step 11

Rewrite $\frac{\frac{y^{2}}{256 x^{4}}}{\frac{y^{4}}{32 x^{7}}}$ as a product.

$\frac{d}{d x} [\frac{y^{2}}{256 x^{4}} \cdot \frac{1}{\frac{y^{4}}{32 x^{7}}}]$

Step 12

Multiply by the reciprocal of the fraction to divide by $\frac{y^{4}}{32 x^{7}}$ .

$\frac{d}{d x} [\frac{y^{2}}{256 x^{4}} (1 \frac{32 x^{7}}{y^{4}})]$

Step 13

Simplify terms.

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Step 13.1

Multiply $\frac{32 x^{7}}{y^{4}}$ by $1$ .

$\frac{d}{d x} [\frac{y^{2}}{256 x^{4}} \cdot \frac{32 x^{7}}{y^{4}}]$

Step 13.2

Multiply $\frac{y^{2}}{256 x^{4}}$ by $\frac{32 x^{7}}{y^{4}}$ .

$\frac{d}{d x} [\frac{y^{2} (32 x^{7})}{256 x^{4} y^{4}}]$

Step 13.3

Move $32$ to the left of $y^{2}$ .

$\frac{d}{d x} [\frac{32 \cdot y^{2} x^{7}}{256 x^{4} y^{4}}]$

Step 13.4

Cancel the common factor of $32$ and $256$ .

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Step 13.4.1

Factor $32$ out of $32 y^{2} x^{7}$ .

$\frac{d}{d x} [\frac{32 (y^{2} x^{7})}{256 x^{4} y^{4}}]$

Step 13.4.2

Cancel the common factors.

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Step 13.4.2.1

Factor $32$ out of $256 x^{4} y^{4}$ .

$\frac{d}{d x} [\frac{32 (y^{2} x^{7})}{32 (8 x^{4} y^{4})}]$

Step 13.4.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{32 (y^{2} x^{7})}{32 (8 x^{4} y^{4})}]$

Step 13.4.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{y^{2} x^{7}}{8 x^{4} y^{4}}]$

Step 13.5

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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Step 13.5.1

Factor $y^{2}$ out of $y^{2} x^{7}$ .

$\frac{d}{d x} [\frac{y^{2} (x^{7})}{8 x^{4} y^{4}}]$

Step 13.5.2

Cancel the common factors.

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Step 13.5.2.1

Factor $y^{2}$ out of $8 x^{4} y^{4}$ .

$\frac{d}{d x} [\frac{y^{2} (x^{7})}{y^{2} (8 x^{4} y^{2})}]$

Step 13.5.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{y^{2} x^{7}}{y^{2} (8 x^{4} y^{2})}]$

Step 13.5.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{7}}{8 x^{4} y^{2}}]$

Step 13.6

Cancel the common factor of $x^{7}$ and $x^{4}$ .

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Step 13.6.1

Factor $x^{4}$ out of $x^{7}$ .

$\frac{d}{d x} [\frac{x^{4} x^{3}}{8 x^{4} y^{2}}]$

Step 13.6.2

Cancel the common factors.

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Step 13.6.2.1

Factor $x^{4}$ out of $8 x^{4} y^{2}$ .

$\frac{d}{d x} [\frac{x^{4} x^{3}}{x^{4} (8 y^{2})}]$

Step 13.6.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x^{4} x^{3}}{x^{4} (8 y^{2})}]$

Step 13.6.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{3}}{8 y^{2}}]$

Step 14

Since $\frac{1}{8 y^{2}}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{8 y^{2}}$ with respect to $x$ is $\frac{1}{8 y^{2}} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{8 y^{2}} \frac{d}{d x} [x^{3}]$

Step 15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 16

Combine fractions.

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Step 16.1

Combine $3$ and $\frac{1}{8 y^{2}}$ .

$\frac{3}{8 y^{2}} x^{2}$

Step 16.2

Combine $\frac{3}{8 y^{2}}$ and $x^{2}$ .

$\frac{3 x^{2}}{8 y^{2}}$