Calculus Examples

x^{3} + 3 x^{2} y + y^{3} = 8

$x^{3} + 3 x^{2} y + y^{3} = 8$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (x^{3} + 3 x^{2} y + y^{3}) = \frac{d}{d x} (8)

$\frac{d}{d x} (x^{3} + 3 x^{2} y + y^{3}) = \frac{d}{d x} (8)$

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

$x^{3} + 3 x^{2} y + y^{3}$ with respect to

x

$x$ is

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

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

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

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

Step 2.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

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

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

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

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

Step 2.2

Evaluate

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

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

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Step 2.2.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3 x^{2} y

$3 x^{2} y$ with respect to

x

$x$ is

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

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

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

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

Step 2.2.2

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = y

$g (x) = y$ .

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

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

Step 2.2.3

Rewrite

\frac{d}{d x} [y]

$\frac{d}{d x} [y]$ as

$y'$ .

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

Step 2.2.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

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

Step 2.2.5

Move

$2$ to the left of

$y$ .

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

Step 2.3

Evaluate

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

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Step 2.3.1

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 2.3.1.1

To apply the Chain Rule, set

$u$ as

$y$ .

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

Step 2.3.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 3$ .

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

Step 2.3.1.3

Replace all occurrences of

$u$ with

$y$ .

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

Step 2.3.2

Rewrite

$\frac{d}{d x} [y]$ as

$y'$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply

$2$ by

$3$ .

$3 x^{2} + 3 x^{2} y' + 6 y x + 3 y^{2} y'$

Step 2.4.3

Reorder terms.

$3 x^{2} + 3 x^{2} y' + 6 x y + 3 y^{2} y'$

Step 3

Since

$8$ is constant with respect to

$x$ , the derivative of

$8$ with respect to

$x$ is

$0$ .

$0$

Step 4

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

$3 x^{2} + 3 x^{2} y' + 6 x y + 3 y^{2} y' = 0$

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

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

$3 x^{2} y' + 6 x y + 3 y^{2} y' = - 3 x^{2}$

Step 5.1.2

Subtract

$6 x y$ from both sides of the equation.

$3 x^{2} y' + 3 y^{2} y' = - 3 x^{2} - 6 x y$

Step 5.2

Factor

$3 y'$ out of

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

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Step 5.2.1

Factor

$3 y'$ out of

$3 x^{2} y'$ .

$3 y' x^{2} + 3 y^{2} y' = - 3 x^{2} - 6 x y$

Step 5.2.2

Factor

$3 y'$ out of

$3 y^{2} y'$ .

$3 y' x^{2} + 3 y' y^{2} = - 3 x^{2} - 6 x y$

Step 5.2.3

Factor

$3 y'$ out of

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

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

Step 5.3

Divide each term in

$3 y' (x^{2} + y^{2}) = - 3 x^{2} - 6 x y$ by

$3 (x^{2} + y^{2})$ and simplify.

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Step 5.3.1

Divide each term in

$3 y' (x^{2} + y^{2}) = - 3 x^{2} - 6 x y$ by

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

$3$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of

$x^{2} + y^{2}$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide

$y'$ by

$1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of

$- 3$ and

$3$ .

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Step 5.3.3.1.1.1

Factor

$3$ out of

$- 3 x^{2}$ .

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.3

Cancel the common factor of

$- 6$ and

$3$ .

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Step 5.3.3.1.3.1

Factor

$3$ out of

$- 6 x y$ .

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

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

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

Step 5.3.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.3.1.4

Move the negative in front of the fraction.

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

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2.2

Factor

$x$ out of

$- x^{2} - 2 x y$ .

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Step 5.3.3.2.2.1

Factor

$x$ out of

$- x^{2}$ .

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

Step 5.3.3.2.2.2

Factor

$x$ out of

$- 2 x y$ .

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

Step 5.3.3.2.2.3

Factor

$x$ out of

$x (- x) + x (- 2 y)$ .

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

Step 5.3.3.2.3

Factor

$- 1$ out of

$- x$ .

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

Step 5.3.3.2.4

Factor

$- 1$ out of

$- 2 y$ .

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

Step 5.3.3.2.5

Factor

$- 1$ out of

$- (x) - (2 y)$ .

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

Step 5.3.3.2.6

Simplify the expression.

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Step 5.3.3.2.6.1

Rewrite

$- (x + 2 y)$ as

$- 1 (x + 2 y)$ .

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

Step 5.3.3.2.6.2

Move the negative in front of the fraction.

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

Step 6

Replace

$y'$ with

$\frac{d y}{d x}$ .

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