Calculus Examples

$y = x + \frac{1}{x}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$x' + \frac{d}{d y} [\frac{1}{x}]$

Step 3.3

Evaluate $\frac{d}{d y} [\frac{1}{x}]$ .

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Step 3.3.1

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 $f (y) = 1$ and $g (y) = x$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

$x' + \frac{x \cdot 0 - 1 \cdot 1 x'}{x^{2}}$

Step 3.3.4

Multiply $x$ by $0$ .

$x' + \frac{0 - 1 \cdot 1 x'}{x^{2}}$

Step 3.3.5

Multiply $- 1$ by $1$ .

$x' + \frac{0 - x'}{x^{2}}$

Step 3.3.6

Subtract $x'$ from $0$ .

$x' + \frac{- x'}{x^{2}}$

Step 3.3.7

Move the negative in front of the fraction.

$x' - \frac{x'}{x^{2}}$

Step 3.4

Reorder terms.

$- \frac{x'}{x^{2}} + x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

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

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

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5.3

Multiply each term in $- \frac{x'}{x^{2}} + x' = 1$ by $x^{2}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $- \frac{x'}{x^{2}} + x' = 1$ by $x^{2}$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.1.1

Move the leading negative in $- \frac{x'}{x^{2}}$ into the numerator.

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

Step 5.3.2.1.2

Cancel the common factor.

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

Step 5.3.2.1.3

Rewrite the expression.

$- x' + x' x^{2} = 1 x^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $x^{2}$ by $1$ .

$- x' + x' x^{2} = x^{2}$

Step 5.4

Solve the equation.

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Step 5.4.1

Factor $x'$ out of $- x' + x' x^{2}$ .

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Step 5.4.1.1

Factor $x'$ out of $- x'$ .

$x' \cdot - 1 + x' x^{2} = x^{2}$

Step 5.4.1.2

Factor $x'$ out of $x' x^{2}$ .

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

Step 5.4.1.3

Factor $x'$ out of $x' \cdot - 1 + x' (x^{2})$ .

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Factor.

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Step 5.4.4.1

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

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

Step 5.4.4.2

Remove unnecessary parentheses.

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

Step 5.4.5

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

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Step 5.4.5.1

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

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

Step 5.4.5.2

Simplify the left side.

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Step 5.4.5.2.1

Cancel the common factor of $x + 1$ .

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Step 5.4.5.2.1.1

Cancel the common factor.

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

Step 5.4.5.2.1.2

Rewrite the expression.

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

Step 5.4.5.2.2

Cancel the common factor of $x - 1$ .

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Step 5.4.5.2.2.1

Cancel the common factor.

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

Step 5.4.5.2.2.2

Divide $x'$ by $1$ .

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

Step 6

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

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