Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.1.2

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

$2 u \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2

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

$2 y y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Rewrite $\frac{1}{1 - x^{2}}$ as ${(1 - x^{2})}^{- 1}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply.

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Step 3.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.5.2

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

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

Step 3.3.6

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

${(1 - x^{2})}^{- 2} (2 x)$

Step 3.3.7

Move $2$ to the left of ${(1 - x^{2})}^{- 2}$ .

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

Step 3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Combine terms.

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Step 3.5.1.1

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

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

Step 3.5.1.2

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

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

Step 3.5.2

Reorder terms.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

Simplify terms.

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Step 5.3.2.1

Combine.

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

Step 5.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.2.3

Multiply $x$ by $1$ .

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

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.3.3.2

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 5.3.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 5.3.3.4

Apply the product rule to $(1 + x) (1 - x)$ .

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

Step 5.3.4

Reorder factors in $\frac{x}{{(1 + x)}^{2} {(1 - x)}^{2} y}$ .

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

Step 6

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

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