Calculus Examples

$\frac{x y}{x^{2} + y^{2}}$

Step 1

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $x^{2} + y^{2}$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2 x$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 9

Combine $y$ and $\frac{- x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}}$ .

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Simplify each term.

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Step 10.2.1

Rewrite using the commutative property of multiplication.

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

Step 10.2.2

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

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Step 10.2.2.1

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

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Step 10.2.2.1.1

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

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

Step 10.2.2.1.2

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

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

Step 10.2.2.2

Add $1$ and $2$ .

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

Step 10.3

Reorder terms.

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

Step 10.4

Simplify the numerator.

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Step 10.4.1

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

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Step 10.4.1.1

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

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

Step 10.4.1.2

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

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

Step 10.4.1.3

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

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

Step 10.4.2

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

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