Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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 $f (x) = x - 1$ and $g (x) = x + 1$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

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

Step 3.2.4.2

Multiply $x + 1$ by $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $1$ and $0$ .

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

Step 3.2.8.2

Multiply $- 1$ by $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Combine the opposite terms in $x + 1 - x - - 1$ .

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Step 3.3.2.1.1

Subtract $x$ from $x$ .

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

Step 3.3.2.1.2

Add $0$ and $1$ .

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

Step 3.3.2.2

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2.3

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

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.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

Combine.

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

Step 5.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

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

Step 5.3.4

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

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

Step 6

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

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