Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = y^{3}$ .

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

Step 3.2

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 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$3 x^{3} y^{2} y' + y^{3} (3 x^{2})$

Step 3.6

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Factor $3 y^{2} y'$ out of $3 y^{2} y' \cdot 1 + 3 y^{2} y' (- x^{3})$ .

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

Step 5.4

Rewrite $1$ as $1^{3}$ .

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

Step 5.5

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

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

Step 5.6

Factor.

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Step 5.6.1

Simplify.

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Step 5.6.1.1

One to any power is one.

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

Step 5.6.1.2

Multiply $x$ by $1$ .

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

Step 5.6.2

Remove unnecessary parentheses.

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

Simplify the left side.

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Step 5.7.2.1

Cancel the common factor of $3$ .

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Step 5.7.2.1.1

Cancel the common factor.

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

Step 5.7.2.1.2

Rewrite the expression.

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

Step 5.7.2.2

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

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Step 5.7.2.2.1

Cancel the common factor.

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

Step 5.7.2.2.2

Rewrite the expression.

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

Step 5.7.2.3

Cancel the common factor of $1 - x$ .

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Step 5.7.2.3.1

Cancel the common factor.

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

Step 5.7.2.3.2

Rewrite the expression.

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

Step 5.7.2.4

Cancel the common factor of $1 + x + x^{2}$ .

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Step 5.7.2.4.1

Cancel the common factor.

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

Step 5.7.2.4.2

Divide $y'$ by $1$ .

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

Step 5.7.3

Simplify the right side.

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Step 5.7.3.1

Simplify each term.

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Step 5.7.3.1.1

Cancel the common factor of $3$ .

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Step 5.7.3.1.1.1

Cancel the common factor.

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

Step 5.7.3.1.1.2

Rewrite the expression.

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

Step 5.7.3.1.2

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

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Step 5.7.3.1.2.1

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

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

Step 5.7.3.1.2.2

Cancel the common factors.

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Step 5.7.3.1.2.2.1

Factor $y^{2}$ out of $(y^{2} (1 - x)) (1 + x + x^{2})$ .

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

Step 5.7.3.1.2.2.2

Cancel the common factor.

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

Step 5.7.3.1.2.2.3

Rewrite the expression.

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

Step 5.7.3.1.3

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

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Step 5.7.3.1.3.1

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

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

Step 5.7.3.1.3.2

Cancel the common factors.

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Step 5.7.3.1.3.2.1

Factor $3$ out of $3 y^{2} (1 - x) (1 + x + x^{2})$ .

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

Step 5.7.3.1.3.2.2

Cancel the common factor.

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

Step 5.7.3.1.3.2.3

Rewrite the expression.

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

Step 5.7.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

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