Calculus Examples

$x = \sqrt{1 - y^{2}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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 $g (x) = 1 - y^{2}$ .

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Step 4.1.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

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

Step 4.5.2

Subtract $2$ from $1$ .

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

Step 4.6

Differentiate.

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Step 4.6.1

Move the negative in front of the fraction.

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

Step 4.6.2

Combine fractions.

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Step 4.6.2.1

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

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

Step 4.6.2.2

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

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

Step 4.6.3

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

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

Step 4.6.4

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

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

Step 4.6.5

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

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

Step 4.6.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

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

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 $y$ .

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

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 $n = 2$ .

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

Step 4.7.3

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

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

Step 4.8

Simplify terms.

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Step 4.8.1

Multiply $2$ by $- 1$ .

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

Step 4.8.2

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

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

Step 4.8.3

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

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

Step 4.8.4

Move $- 2$ to the left of $y$ .

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

Step 4.8.5

Factor $2$ out of $- 2 y$ .

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

Step 4.9

Cancel the common factors.

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Step 4.9.1

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

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

Step 4.9.2

Cancel the common factor.

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

Step 4.9.3

Rewrite the expression.

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

Step 4.10

Move the negative in front of the fraction.

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Reorder factors in $y' y$ .

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

Step 5

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

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

Step 6

Solve for $y'$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

Divide $\frac{y y'}{{(1 - y^{2})}^{\frac{1}{2}}}$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $1$ by $- 1$ .

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

Step 6.3

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

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

Step 6.4

Simplify the left side.

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Step 6.4.1

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

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Step 6.4.1.1

Cancel the common factor.

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

Step 6.4.1.2

Rewrite the expression.

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $y$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $y'$ by $1$ .

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

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Move the negative in front of the fraction.

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

Step 7

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

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