Calculus Examples

$3 {(x^{2} + y^{2})}^{2} = 100 x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (100 x y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $3 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + y^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

$3 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$3 (2 u_{1} \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $2$ by $3$ .

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

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

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

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

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

Step 2.4.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder the factors of $(6 x^{2} + 6 y^{2}) (2 x + 2 y y')$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $100$ is constant with respect to $x$ , the derivative of $100 x y$ with respect to $x$ is $100 \frac{d}{d x} [x y]$ .

$100 \frac{d}{d x} [x y]$

Step 3.2

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 $f (x) = x$ and $g (x) = y$ .

$100 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.3

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

$100 (x y' + y \frac{d}{d x} [x])$

Step 3.4

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

$100 (x y' + y \cdot 1)$

Step 3.5

Multiply $y$ by $1$ .

$100 (x y' + y)$

Step 3.6

Apply the distributive property.

$100 x y' + 100 y$

Step 4

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

$(2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 x y' + 100 y$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $(2 x + 2 y y') (6 x^{2} + 6 y^{2})$ .

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Step 5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 x y' + 100 y$

Step 5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 x y' + 100 y$

Step 5.1.3

Expand $(2 x + 2 y y') (6 x^{2} + 6 y^{2})$ using the FOIL Method.

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Step 5.1.3.1

Apply the distributive property.

$2 x (6 x^{2} + 6 y^{2}) + 2 y y' (6 x^{2} + 6 y^{2}) = 100 x y' + 100 y$

Step 5.1.3.2

Apply the distributive property.

$2 x (6 x^{2}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2} + 6 y^{2}) = 100 x y' + 100 y$

Step 5.1.3.3

Apply the distributive property.

$2 x (6 x^{2}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 6 x \cdot x^{2} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1.4.2.1

Move $x^{2}$ .

$2 \cdot 6 (x^{2} x) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.2.2

Multiply $x^{2}$ by $x$ .

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Step 5.1.4.2.2.1

Raise $x$ to the power of $1$ .

$2 \cdot 6 (x^{2} x^{1}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \cdot 6 x^{2 + 1} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 6 x^{3} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.3

Multiply $2$ by $6$ .

$12 x^{3} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.4

Rewrite using the commutative property of multiplication.

$12 x^{3} + 2 \cdot 6 x y^{2} + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.5

Multiply $2$ by $6$ .

$12 x^{3} + 12 x y^{2} + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.6

Multiply $6$ by $2$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y y' (6 y^{2}) = 100 x y' + 100 y$

Step 5.1.4.7

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 5.1.4.7.1

Move $y^{2}$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 (y^{2} y) y' \cdot 6 = 100 x y' + 100 y$

Step 5.1.4.7.2

Multiply $y^{2}$ by $y$ .

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Step 5.1.4.7.2.1

Raise $y$ to the power of $1$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 6 = 100 x y' + 100 y$

Step 5.1.4.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y^{2 + 1} y' \cdot 6 = 100 x y' + 100 y$

Step 5.1.4.7.3

Add $2$ and $1$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y^{3} y' \cdot 6 = 100 x y' + 100 y$

Step 5.1.4.8

Multiply $6$ by $2$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' = 100 x y' + 100 y$

Step 5.2

Subtract $100 x y'$ from both sides of the equation.

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' - 100 x y' = 100 y$

Step 5.3

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

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Step 5.3.1

Subtract $12 x^{3}$ from both sides of the equation.

$12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' - 100 x y' = 100 y - 12 x^{3}$

Step 5.3.2

Subtract $12 x y^{2}$ from both sides of the equation.

$12 y y' x^{2} + 12 y^{3} y' - 100 x y' = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.4

Factor $4 y'$ out of $12 y y' x^{2} + 12 y^{3} y' - 100 x y'$ .

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Step 5.4.1

Factor $4 y'$ out of $12 y y' x^{2}$ .

$4 y' (3 y x^{2}) + 12 y^{3} y' - 100 x y' = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.4.2

Factor $4 y'$ out of $12 y^{3} y'$ .

$4 y' (3 y x^{2}) + 4 y' (3 y^{3}) - 100 x y' = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.4.3

Factor $4 y'$ out of $- 100 x y'$ .

$4 y' (3 y x^{2}) + 4 y' (3 y^{3}) + 4 y' (- 25 x) = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.4.4

Factor $4 y'$ out of $4 y' (3 y x^{2}) + 4 y' (3 y^{3})$ .

$4 y' (3 y x^{2} + 3 y^{3}) + 4 y' (- 25 x) = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.4.5

Factor $4 y'$ out of $4 y' (3 y x^{2} + 3 y^{3}) + 4 y' (- 25 x)$ .

$4 y' (3 y x^{2} + 3 y^{3} - 25 x) = 100 y - 12 x^{3} - 12 x y^{2}$

Step 5.5

Divide each term in $4 y' (3 y x^{2} + 3 y^{3} - 25 x) = 100 y - 12 x^{3} - 12 x y^{2}$ by $4 (3 y x^{2} + 3 y^{3} - 25 x)$ and simplify.

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Step 5.5.1

Divide each term in $4 y' (3 y x^{2} + 3 y^{3} - 25 x) = 100 y - 12 x^{3} - 12 x y^{2}$ by $4 (3 y x^{2} + 3 y^{3} - 25 x)$ .

$\frac{4 y' (3 y x^{2} + 3 y^{3} - 25 x)}{4 (3 y x^{2} + 3 y^{3} - 25 x)} = \frac{100 y}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $4$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y' (3 y x^{2} + 3 y^{3} - 25 x)}{3 y x^{2} + 3 y^{3} - 25 x} = \frac{100 y}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.2.2

Cancel the common factor of $3 y x^{2} + 3 y^{3} - 25 x$ .

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{100 y}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $100$ and $4$ .

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Step 5.5.3.1.1.1

Factor $4$ out of $100 y$ .

$y' = \frac{4 (25 y)}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Cancel the common factor.

$y' = \frac{4 (25 y)}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.1.2.2

Rewrite the expression.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{- 12 x^{3}}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.2

Cancel the common factor of $- 12$ and $4$ .

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Step 5.5.3.1.2.1

Factor $4$ out of $- 12 x^{3}$ .

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{4 (- 3 x^{3})}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Cancel the common factor.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{4 (- 3 x^{3})}{4 (3 y x^{2} + 3 y^{3} - 25 x)} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.2.2.2

Rewrite the expression.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{- 3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{- 12 x y^{2}}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.4

Cancel the common factor of $- 12$ and $4$ .

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Step 5.5.3.1.4.1

Factor $4$ out of $- 12 x y^{2}$ .

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{4 (- 3 x y^{2})}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.4.2

Cancel the common factors.

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Step 5.5.3.1.4.2.1

Cancel the common factor.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{4 (- 3 x y^{2})}{4 (3 y x^{2} + 3 y^{3} - 25 x)}$

Step 5.5.3.1.4.2.2

Rewrite the expression.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} + \frac{- 3 x y^{2}}{3 y x^{2} + 3 y^{3} - 25 x}$

Step 5.5.3.1.5

Move the negative in front of the fraction.

$y' = \frac{25 y}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x y^{2}}{3 y x^{2} + 3 y^{3} - 25 x}$

Step 5.5.3.2

Combine into one fraction.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{25 y - 3 x^{3}}{3 y x^{2} + 3 y^{3} - 25 x} - \frac{3 x y^{2}}{3 y x^{2} + 3 y^{3} - 25 x}$

Step 5.5.3.2.2

Combine the numerators over the common denominator.

$y' = \frac{25 y - 3 x^{3} - 3 x y^{2}}{3 y x^{2} + 3 y^{3} - 25 x}$

Step 6

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

$\frac{d y}{d x} = \frac{25 y - 3 x^{3} - 3 x y^{2}}{3 y x^{2} + 3 y^{3} - 25 x}$