Calculus Examples

$x^{3} y^{3} + x = 9$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} y^{3} + x) = \frac{d}{d x} (9)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} y^{3} + x$ with respect to $x$ is $\frac{d}{d x} [x^{3} y^{3}] + \frac{d}{d x} [x]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [x^{3} y^{3}]$ .

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Step 2.2.1

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

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

Step 2.2.2

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 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Move $3$ to the left of $x^{3}$ .

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

Step 2.2.6

Move $3$ to the left of $y^{3}$ .

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

Step 2.3

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

$3 x^{3} y^{2} y' + 3 y^{3} x^{2} + 1$

Step 3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0$

Step 4

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

$3 x^{3} y^{2} y' + 3 y^{3} x^{2} + 1 = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1.1

Subtract $3 y^{3} x^{2}$ from both sides of the equation.

$3 x^{3} y^{2} y' + 1 = - 3 y^{3} x^{2}$

Step 5.1.2

Subtract $1$ from both sides of the equation.

$3 x^{3} y^{2} y' = - 3 y^{3} x^{2} - 1$

Step 5.2

Divide each term in $3 x^{3} y^{2} y' = - 3 y^{3} x^{2} - 1$ by $3 x^{3} y^{2}$ and simplify.

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Step 5.2.1

Divide each term in $3 x^{3} y^{2} y' = - 3 y^{3} x^{2} - 1$ by $3 x^{3} y^{2}$ .

$\frac{3 x^{3} y^{2} y'}{3 x^{3} y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x^{3} y^{2} y'}{3 x^{3} y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{x^{3} y^{2} y'}{x^{3} y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2.2

Cancel the common factor of $x^{3}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{x^{3} y^{2} y'}{x^{3} y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{y^{2} y'}{y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2.3

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

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{y^{2} y'}{y^{2}} = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 y^{3} x^{2}}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 5.2.3.1.1.1

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

$y' = \frac{3 (- y^{3} x^{2})}{3 x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

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

$y' = \frac{3 (- y^{3} x^{2})}{3 (x^{3} y^{2})} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$y' = \frac{3 (- y^{3} x^{2})}{3 (x^{3} y^{2})} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- y^{3} x^{2}}{x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.2

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 5.2.3.1.2.1

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

$y' = \frac{y^{2} (- y x^{2})}{x^{3} y^{2}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

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

$y' = \frac{y^{2} (- y x^{2})}{y^{2} x^{3}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.2.2.2

Cancel the common factor.

$y' = \frac{y^{2} (- y x^{2})}{y^{2} x^{3}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y' = \frac{- y x^{2}}{x^{3}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.2.3.1.3.1

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

$y' = \frac{x^{2} (- y)}{x^{3}} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

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

$y' = \frac{x^{2} (- y)}{x^{2} x} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.3.2.2

Cancel the common factor.

$y' = \frac{x^{2} (- y)}{x^{2} x} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.3.2.3

Rewrite the expression.

$y' = \frac{- y}{x} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{y}{x} + \frac{- 1}{3 x^{3} y^{2}}$

Step 5.2.3.1.5

Move the negative in front of the fraction.

$y' = - \frac{y}{x} - \frac{1}{3 x^{3} y^{2}}$

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x} - \frac{1}{3 x^{3} y^{2}}$