Calculus Examples

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{{(- 4 x y + 1)}^{3}} = 0$

Step 1

Set the numerator equal to zero.

$4 (- 32 y^{4} x + 12 y^{3}) = 0$

Step 2

Solve the equation for $x$ .

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Step 2.1

Divide each term in $4 (- 32 y^{4} x + 12 y^{3}) = 0$ by $4$ and simplify.

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Step 2.1.1

Divide each term in $4 (- 32 y^{4} x + 12 y^{3}) = 0$ by $4$ .

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{4} = \frac{0}{4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{4 (- 32 y^{4} x + 12 y^{3})}{4} = \frac{0}{4}$

Step 2.1.2.1.2

Divide $- 32 y^{4} x + 12 y^{3}$ by $1$ .

$- 32 y^{4} x + 12 y^{3} = \frac{0}{4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $4$ .

$- 32 y^{4} x + 12 y^{3} = 0$

Step 2.2

Subtract $12 y^{3}$ from both sides of the equation.

$- 32 y^{4} x = - 12 y^{3}$

Step 2.3

Divide each term in $- 32 y^{4} x = - 12 y^{3}$ by $- 32 y^{4}$ and simplify.

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Step 2.3.1

Divide each term in $- 32 y^{4} x = - 12 y^{3}$ by $- 32 y^{4}$ .

$\frac{- 32 y^{4} x}{- 32 y^{4}} = \frac{- 12 y^{3}}{- 32 y^{4}}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 32$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 32 y^{4} x}{- 32 y^{4}} = \frac{- 12 y^{3}}{- 32 y^{4}}$

Step 2.3.2.1.2

Rewrite the expression.

$\frac{y^{4} x}{y^{4}} = \frac{- 12 y^{3}}{- 32 y^{4}}$

Step 2.3.2.2

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

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{y^{4} x}{y^{4}} = \frac{- 12 y^{3}}{- 32 y^{4}}$

Step 2.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 12 y^{3}}{- 32 y^{4}}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 12$ and $- 32$ .

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Step 2.3.3.1.1

Factor $- 4$ out of $- 12 y^{3}$ .

$x = \frac{- 4 (3 y^{3})}{- 32 y^{4}}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $- 4$ out of $- 32 y^{4}$ .

$x = \frac{- 4 (3 y^{3})}{- 4 (8 y^{4})}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 (3 y^{3})}{- 4 (8 y^{4})}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x = \frac{3 y^{3}}{8 y^{4}}$

Step 2.3.3.2

Cancel the common factor of $y^{3}$ and $y^{4}$ .

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Step 2.3.3.2.1

Factor $y^{3}$ out of $3 y^{3}$ .

$x = \frac{y^{3} \cdot 3}{8 y^{4}}$

Step 2.3.3.2.2

Cancel the common factors.

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Step 2.3.3.2.2.1

Factor $y^{3}$ out of $8 y^{4}$ .

$x = \frac{y^{3} \cdot 3}{y^{3} (8 y)}$

Step 2.3.3.2.2.2

Cancel the common factor.

$x = \frac{y^{3} \cdot 3}{y^{3} (8 y)}$

Step 2.3.3.2.2.3

Rewrite the expression.

$x = \frac{3}{8 y}$

Step 3

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = \frac{3}{8 y}$