Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $z$ with respect to $x$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{3} y - x y^{3}$ and $g (x) = x^{2} + y^{2}$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Move $3$ to the left of $y$ .

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Multiply $- 1$ by $1$ .

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$\frac{(x^{2} + y^{2}) (3 y x^{2} - y^{3}) - (x^{3} y - x y^{3}) (2 x + 0)}{{(x^{2} + y^{2})}^{2}}$

Step 3.2.11

Simplify the expression.

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Step 3.2.11.1

Add $2 x$ and $0$ .

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

Step 3.2.11.2

Multiply $2$ by $- 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Expand $(x^{2} + y^{2}) (3 y x^{2} - y^{3})$ using the FOIL Method.

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Step 3.3.3.1.1.1

Apply the distributive property.

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

Step 3.3.3.1.1.2

Apply the distributive property.

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

Step 3.3.3.1.1.3

Apply the distributive property.

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

Step 3.3.3.1.2

Simplify and combine like terms.

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Step 3.3.3.1.2.1

Simplify each term.

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Step 3.3.3.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.2.1.2

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

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Step 3.3.3.1.2.1.2.1

Move $x^{2}$ .

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

Step 3.3.3.1.2.1.2.2

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

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

Step 3.3.3.1.2.1.2.3

Add $2$ and $2$ .

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

Step 3.3.3.1.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.2.1.5

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

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Step 3.3.3.1.2.1.5.1

Move $y$ .

$\frac{3 x^{4} y - x^{2} y^{3} + 3 (y \cdot y^{2}) x^{2} + y^{2} (- y^{3}) - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.2.1.5.2

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

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Step 3.3.3.1.2.1.5.2.1

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

$\frac{3 x^{4} y - x^{2} y^{3} + 3 (y^{1} y^{2}) x^{2} + y^{2} (- y^{3}) - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.2.1.5.2.2

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

$\frac{3 x^{4} y - x^{2} y^{3} + 3 y^{1 + 2} x^{2} + y^{2} (- y^{3}) - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.2.1.5.3

Add $1$ and $2$ .

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

Step 3.3.3.1.2.1.6

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.2.1.7

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 3.3.3.1.2.1.7.1

Move $y^{3}$ .

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

Step 3.3.3.1.2.1.7.2

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

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

Step 3.3.3.1.2.1.7.3

Add $3$ and $2$ .

$\frac{3 x^{4} y - x^{2} y^{3} + 3 y^{3} x^{2} - y^{5} - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.2.2

Add $- x^{2} y^{3}$ and $3 y^{3} x^{2}$ .

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Step 3.3.3.1.2.2.1

Move $y^{3}$ .

$\frac{3 x^{4} y - x^{2} y^{3} + 3 x^{2} y^{3} - y^{5} - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.2.2.2

Add $- x^{2} y^{3}$ and $3 x^{2} y^{3}$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 (x^{3} y) x - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.3.3.1.3.1

Move $x$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 (x \cdot x^{3} y) - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.3.2

Multiply $x$ by $x^{3}$ .

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Step 3.3.3.1.3.2.1

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

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 (x^{1} x^{3} y) - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.3.2.2

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

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 (x^{1 + 3} y) - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.3.3

Add $1$ and $3$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 (x^{4} y) - 2 (- x y^{3}) x}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.3.1.4.1

Move $x$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 x^{4} y - 2 (- (x \cdot x) y^{3})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.4.2

Multiply $x$ by $x$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 x^{4} y - 2 (- x^{2} y^{3})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.1.5

Multiply $- 1$ by $- 2$ .

$\frac{3 x^{4} y + 2 x^{2} y^{3} - y^{5} - 2 x^{4} y + 2 x^{2} y^{3}}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.2

Subtract $2 x^{4} y$ from $3 x^{4} y$ .

$\frac{2 x^{2} y^{3} - y^{5} + 1 x^{4} y + 2 x^{2} y^{3}}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.3

Multiply $x^{4}$ by $1$ .

$\frac{2 x^{2} y^{3} - y^{5} + x^{4} y + 2 x^{2} y^{3}}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.3.4

Add $2 x^{2} y^{3}$ and $2 x^{2} y^{3}$ .

$\frac{- y^{5} + x^{4} y + 4 x^{2} y^{3}}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.4

Reorder terms.

$\frac{- y^{5} + x^{4} y + 4 y^{3} x^{2}}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.5

Factor $y$ out of $- y^{5} + x^{4} y + 4 y^{3} x^{2}$ .

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Step 3.3.5.1

Factor $y$ out of $- y^{5}$ .

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

Step 3.3.5.2

Factor $y$ out of $x^{4} y$ .

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

Step 3.3.5.3

Factor $y$ out of $4 y^{3} x^{2}$ .

$\frac{y (- y^{4}) + y x^{4} + y (4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.5.4

Factor $y$ out of $y (- y^{4}) + y x^{4}$ .

$\frac{y (- y^{4} + x^{4}) + y (4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.5.5

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

$\frac{y (- y^{4} + x^{4} + 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.6

Factor $- 1$ out of $- y^{4}$ .

$\frac{y (- (y^{4}) + x^{4} + 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.7

Factor $- 1$ out of $x^{4}$ .

$\frac{y (- (y^{4}) - 1 (- x^{4}) + 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.8

Factor $- 1$ out of $- (y^{4}) - 1 (- x^{4})$ .

$\frac{y (- (y^{4} - x^{4}) + 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.9

Factor $- 1$ out of $4 y^{2} x^{2}$ .

$\frac{y (- (y^{4} - x^{4}) - (- 4 y^{2} x^{2}))}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.10

Factor $- 1$ out of $- (y^{4} - x^{4}) - (- 4 y^{2} x^{2})$ .

$\frac{y (- (y^{4} - x^{4} - 4 y^{2} x^{2}))}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.11

Rewrite $- (y^{4} - x^{4} - 4 y^{2} x^{2})$ as $- 1 (y^{4} - x^{4} - 4 y^{2} x^{2})$ .

$\frac{y (- 1 (y^{4} - x^{4} - 4 y^{2} x^{2}))}{{(x^{2} + y^{2})}^{2}}$

Step 3.3.12

Move the negative in front of the fraction.

$- \frac{y (y^{4} - x^{4} - 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$

Step 4

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

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

Step 5

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

$\frac{d z}{d x} = - \frac{y (y^{4} - x^{4} - 4 y^{2} x^{2})}{{(x^{2} + y^{2})}^{2}}$