Calculus Examples

$\frac{x}{y^{3}} = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\frac{x}{y^{3}}) = \frac{d}{d y} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x$ and $g (y) = y^{3}$ .

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

Step 2.2

Multiply the exponents in ${(y^{3})}^{2}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $2$ .

$\frac{y^{3} \frac{d}{d y} [x] - x \frac{d}{d y} [y^{3}]}{y^{6}}$

Step 2.3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{y^{3} x' - x \frac{d}{d y} [y^{3}]}{y^{6}}$

Step 2.4

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

$\frac{y^{3} x' - x (3 y^{2})}{y^{6}}$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $3$ by $- 1$ .

$\frac{y^{3} x' - 3 x y^{2}}{y^{6}}$

Step 2.5.2

Factor $y^{2}$ out of $y^{3} x' - 3 x y^{2}$ .

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Step 2.5.2.1

Factor $y^{2}$ out of $y^{3} x'$ .

$\frac{y^{2} (y x') - 3 x y^{2}}{y^{6}}$

Step 2.5.2.2

Factor $y^{2}$ out of $- 3 x y^{2}$ .

$\frac{y^{2} (y x') + y^{2} (- 3 x)}{y^{6}}$

Step 2.5.2.3

Factor $y^{2}$ out of $y^{2} (y x') + y^{2} (- 3 x)$ .

$\frac{y^{2} (y x' - 3 x)}{y^{6}}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $y^{2}$ out of $y^{6}$ .

$\frac{y^{2} (y x' - 3 x)}{y^{2} y^{4}}$

Step 2.6.2

Cancel the common factor.

$\frac{y^{2} (y x' - 3 x)}{y^{2} y^{4}}$

Step 2.6.3

Rewrite the expression.

$\frac{y x' - 3 x}{y^{4}}$

Step 3

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$\frac{y x' - 3 x}{y^{4}} = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Set the numerator equal to zero.

$y x' - 3 x = 0$

Step 5.2

Solve the equation for $x'$ .

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Step 5.2.1

Add $3 x$ to both sides of the equation.

$y x' = 3 x$

Step 5.2.2

Divide each term in $y x' = 3 x$ by $y$ and simplify.

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Step 5.2.2.1

Divide each term in $y x' = 3 x$ by $y$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $y$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x'$ by $1$ .

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

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

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