Calculus Examples

$x = \sqrt{x^{2} + y^{2}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

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

Step 4.5.2

Subtract $2$ from $1$ .

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

Step 4.6

Differentiate.

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Step 4.6.1

Move the negative in front of the fraction.

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

Step 4.6.2

Combine fractions.

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Step 4.6.2.1

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

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

Step 4.6.2.2

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

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

Step 4.6.3

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

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

Step 4.6.4

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

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

Step 4.7

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 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.8

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

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

Step 4.9

Simplify.

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Step 4.9.1

Reorder the factors of $\frac{1}{2 {(x^{2} + y^{2})}^{\frac{1}{2}}} (2 x + 2 y y')$ .

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

Step 4.9.2

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

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

Step 4.9.3

Factor $2$ out of $2 x$ .

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

Step 4.9.4

Factor $2$ out of $2 y y'$ .

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

Step 4.9.5

Factor $2$ out of $2 (x) + 2 (y y')$ .

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

Step 4.9.6

Cancel the common factors.

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Step 4.9.6.1

Factor $2$ out of $2 {(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 4.9.6.2

Cancel the common factor.

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

Step 4.9.6.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Solve for $y'$ .

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Step 6.1

Rewrite the equation as $\frac{x + y y'}{{(x^{2} + y^{2})}^{\frac{1}{2}}} = 1$ .

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{x + y y'}{{(x^{2} + y^{2})}^{\frac{1}{2}}} {(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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Step 6.3.1.1.1

Cancel the common factor of ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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Step 6.3.1.1.1.1

Cancel the common factor.

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

Step 6.3.1.1.1.2

Rewrite the expression.

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

Step 6.3.1.1.2

Simplify with commuting.

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Step 6.3.1.1.2.1

Reorder $y$ and $y'$ .

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

Step 6.3.1.1.2.2

Reorder $x$ and $y' y$ .

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

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

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

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

Step 6.4

Solve for $y'$ .

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Step 6.4.1

Subtract $x$ from both sides of the equation.

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

Step 6.4.2

Divide each term in $y' y = {(x^{2} + y^{2})}^{\frac{1}{2}} - x$ by $y$ and simplify.

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Step 6.4.2.1

Divide each term in $y' y = {(x^{2} + y^{2})}^{\frac{1}{2}} - x$ by $y$ .

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

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of $y$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

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

Step 6.4.2.2.1.2

Divide $y'$ by $1$ .

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

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Combine the numerators over the common denominator.

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

Step 7

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

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