Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $3$ by $2$ .

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

Step 2.2.2

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

$\frac{y^{3} \cdot 1 - x \frac{d}{d x} [y^{3}]}{y^{6}}$

Step 2.2.3

Multiply $y^{3}$ by $1$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.3.2

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

$\frac{y^{3} - x (3 u^{2} \frac{d}{d x} [y])}{y^{6}}$

Step 2.3.3

Replace all occurrences of $u$ with $y$ .

$\frac{y^{3} - x (3 y^{2} \frac{d}{d x} [y])}{y^{6}}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Multiply $3$ by $- 1$ .

$\frac{y^{3} - 3 x (y^{2} \frac{d}{d x} [y])}{y^{6}}$

Step 2.4.2

Factor $y^{2}$ out of $y^{3} - 3 x y^{2} \frac{d}{d x} [y]$ .

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Step 2.4.2.1

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

$\frac{y^{2} y - 3 x y^{2} \frac{d}{d x} [y]}{y^{6}}$

Step 2.4.2.2

Factor $y^{2}$ out of $- 3 x y^{2} \frac{d}{d x} [y]$ .

$\frac{y^{2} y + y^{2} (- 3 x \frac{d}{d x} [y])}{y^{6}}$

Step 2.4.2.3

Factor $y^{2}$ out of $y^{2} y + y^{2} (- 3 x \frac{d}{d x} [y])$ .

$\frac{y^{2} (y - 3 x \frac{d}{d x} [y])}{y^{6}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factor.

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

Step 2.5.3

Rewrite the expression.

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

Step 2.6

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

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

Step 2.7

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$- 3 x y' + y = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Subtract $y$ from both sides of the equation.

$- 3 x y' = - y$

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 6

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

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