Calculus Examples

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

Step 1

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

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

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(x^{2} + y^{2})}^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Multiply $2$ by $2$ .

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

Step 3.2

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

$2 y \frac{{(x^{2} + y^{2})}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]}{{(x^{2} + y^{2})}^{4}}$

Step 3.3

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

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

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

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Step 4.1

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

$2 y \frac{{(x^{2} + y^{2})}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + y^{2}])}{{(x^{2} + y^{2})}^{4}}$

Step 4.2

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

$2 y \frac{{(x^{2} + y^{2})}^{2} - x (2 u \frac{d}{d x} [x^{2} + y^{2}])}{{(x^{2} + y^{2})}^{4}}$

Step 4.3

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

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

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 6

Cancel the common factors.

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the expression.

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Step 10.1

Add $2 x$ and $0$ .

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

Step 10.2

Multiply $2$ by $- 2$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

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

Step 18.2

Apply the distributive property.

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

Step 18.3

Simplify each term.

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Step 18.3.1

Rewrite using the commutative property of multiplication.

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

Step 18.3.2

Multiply $2$ by $- 3$ .

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

Step 18.3.3

Rewrite using the commutative property of multiplication.

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

Step 18.3.4

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

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Step 18.3.4.1

Move $y^{2}$ .

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

Step 18.3.4.2

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

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Step 18.3.4.2.1

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

$\frac{- 6 y x^{2} + 2 (y^{2} y^{1})}{{(x^{2} + y^{2})}^{3}}$

Step 18.3.4.2.2

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

$\frac{- 6 y x^{2} + 2 y^{2 + 1}}{{(x^{2} + y^{2})}^{3}}$

Step 18.3.4.3

Add $2$ and $1$ .

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

Step 18.4

Reorder terms.

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

Step 18.5

Factor $2 y$ out of $2 y^{3} - 6 x^{2} y$ .

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Step 18.5.1

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

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

Step 18.5.2

Factor $2 y$ out of $- 6 x^{2} y$ .

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

Step 18.5.3

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

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