Calculus Examples

$\frac{1 + x + y}{\sqrt{1 + x^{2} + y^{2}}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x^{2} + y^{2}}$ as ${(1 + x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 1 + x + y$ and $g (y) = {(1 + x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3

Multiply the exponents in ${({(1 + x^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $1 + x + y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [x] + \frac{d}{d y} [y]$ .

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

Step 5.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{{(1 + x^{2} + y^{2})}^{\frac{1}{2}} (0 + \frac{d}{d y} [x] + \frac{d}{d y} [y]) - (1 + x + y) \frac{d}{d y} [{(1 + x^{2} + y^{2})}^{\frac{1}{2}}]}{1 + x^{2} + y^{2}}$

Step 5.3

Add $0$ and $\frac{d}{d y} [x]$ .

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

Step 5.4

Since $x$ is constant with respect to $y$ , the derivative of $x$ with respect to $y$ is $0$ .

$\frac{{(1 + x^{2} + y^{2})}^{\frac{1}{2}} (0 + \frac{d}{d y} [y]) - (1 + x + y) \frac{d}{d y} [{(1 + x^{2} + y^{2})}^{\frac{1}{2}}]}{1 + x^{2} + y^{2}}$

Step 5.5

Add $0$ and $\frac{d}{d y} [y]$ .

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

Step 5.6

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

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

Step 5.7

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

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

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{1}{2}}$ and $g (y) = 1 + x^{2} + y^{2}$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

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

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

Step 11.3

Move ${(1 + x^{2} + y^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12

By the Sum Rule, the derivative of $1 + x^{2} + y^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}]$ .

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

Step 13

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{{(1 + x^{2} + y^{2})}^{\frac{1}{2}} - (1 + x + y) (\frac{1}{2 {(1 + x^{2} + y^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}]))}{1 + x^{2} + y^{2}}$

Step 14

Add $0$ and $\frac{d}{d y} [x^{2}]$ .

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

Step 15

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

$\frac{{(1 + x^{2} + y^{2})}^{\frac{1}{2}} - (1 + x + y) (\frac{1}{2 {(1 + x^{2} + y^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d y} [y^{2}]))}{1 + x^{2} + y^{2}}$

Step 16

Add $0$ and $\frac{d}{d y} [y^{2}]$ .

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

Step 17

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

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

Step 18

Simplify terms.

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Step 18.1

Combine $2$ and $\frac{1}{2 {(1 + x^{2} + y^{2})}^{\frac{1}{2}}}$ .

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

Step 18.2

Combine $\frac{2}{2 {(1 + x^{2} + y^{2})}^{\frac{1}{2}}}$ and $y$ .

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

Step 18.3

Cancel the common factor.

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

Step 18.4

Rewrite the expression.

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

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

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

Step 19.2

Simplify the numerator.

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Step 19.2.1

Let $u_{2} = {(1 + x^{2} + y^{2})}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(1 + x^{2} + y^{2})}^{\frac{1}{2}}$ .

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Step 19.2.1.1

Multiply $u_{2}$ by $u_{2}$ .

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

Step 19.2.1.2

Apply the distributive property.

$\frac{\frac{{u_{2}}^{2} - 1 y - x y - y \cdot y}{u_{2}}}{1 + x^{2} + y^{2}}$

Step 19.2.1.3

Simplify.

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Step 19.2.1.3.1

Rewrite $- 1 y$ as $- y$ .

$\frac{\frac{{u_{2}}^{2} - y - x y - y \cdot y}{u_{2}}}{1 + x^{2} + y^{2}}$

Step 19.2.1.3.2

Multiply $y$ by $y$ by adding the exponents.

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Step 19.2.1.3.2.1

Move $y$ .

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

Step 19.2.1.3.2.2

Multiply $y$ by $y$ .

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

Step 19.2.2

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

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

Step 19.2.3

Simplify.

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Step 19.2.3.1

Simplify each term.

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Step 19.2.3.1.1

Multiply the exponents in ${({(1 + x^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 19.2.3.1.1.1

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

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

Step 19.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 19.2.3.1.1.2.1

Cancel the common factor.

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

Step 19.2.3.1.1.2.2

Rewrite the expression.

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

Step 19.2.3.1.2

Simplify.

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

Step 19.2.3.2

Combine the opposite terms in $1 + x^{2} + y^{2} - y - x y - y^{2}$ .

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Step 19.2.3.2.1

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

$\frac{\frac{1 + x^{2} - y - x y + 0}{{(1 + x^{2} + y^{2})}^{\frac{1}{2}}}}{1 + x^{2} + y^{2}}$

Step 19.2.3.2.2

Add $1 + x^{2} - y - x y$ and $0$ .

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

Step 19.3

Combine terms.

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Step 19.3.1

Rewrite $\frac{\frac{1 + x^{2} - y - x y}{{(1 + x^{2} + y^{2})}^{\frac{1}{2}}}}{1 + x^{2} + y^{2}}$ as a product.

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

Step 19.3.2

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

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

Step 19.3.3

Multiply ${(1 + x^{2} + y^{2})}^{\frac{1}{2}}$ by $1 + x^{2} + y^{2}$ by adding the exponents.

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Step 19.3.3.1

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

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Step 19.3.3.1.1

Raise $1 + x^{2} + y^{2}$ to the power of $1$ .

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

Step 19.3.3.1.2

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

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

Step 19.3.3.2

Write $1$ as a fraction with a common denominator.

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

Step 19.3.3.3

Combine the numerators over the common denominator.

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

Step 19.3.3.4

Add $1$ and $2$ .

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

Step 19.4

Reorder terms.

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