Calculus Examples

$\frac{3}{x} - \frac{x}{y} = 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{3}{x} - \frac{x}{y}) = \frac{d}{d x} (2)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $\frac{3}{x} - \frac{x}{y}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{x}] + \frac{d}{d x} [- \frac{x}{y}]$ .

$\frac{d}{d x} [\frac{3}{x}] + \frac{d}{d x} [- \frac{x}{y}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{3}{x}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

$3 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [- \frac{x}{y}]$

Step 2.2.2

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

$3 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [- \frac{x}{y}]$

Step 2.2.3

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

$3 (- x^{- 2}) + \frac{d}{d x} [- \frac{x}{y}]$

Step 2.2.4

Multiply $- 1$ by $3$ .

$- 3 x^{- 2} + \frac{d}{d x} [- \frac{x}{y}]$

Step 2.3

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

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Step 2.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x}{y}$ .

$- 3 x^{- 2} - \frac{d}{d x} [\frac{x}{y}] + \frac{x}{y} \frac{d}{d x} [- 1]$

Step 2.3.2

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

$- 3 x^{- 2} - \frac{y \frac{d}{d x} [x] - x \frac{d}{d x} [y]}{y^{2}} + \frac{x}{y} \frac{d}{d x} [- 1]$

Step 2.3.3

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

$- 3 x^{- 2} - \frac{y \cdot 1 - x \frac{d}{d x} [y]}{y^{2}} + \frac{x}{y} \frac{d}{d x} [- 1]$

Step 2.3.4

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

$- 3 x^{- 2} - \frac{y \cdot 1 - x y'}{y^{2}} + \frac{x}{y} \frac{d}{d x} [- 1]$

Step 2.3.5

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

$- 3 x^{- 2} - \frac{y \cdot 1 - x y'}{y^{2}} + \frac{x}{y} \cdot 0$

Step 2.3.6

Multiply $y$ by $1$ .

$- 3 x^{- 2} - \frac{y - x y'}{y^{2}} + \frac{x}{y} \cdot 0$

Step 2.3.7

Multiply $\frac{x}{y}$ by $0$ .

$- 3 x^{- 2} - \frac{y - x y'}{y^{2}} + 0$

Step 2.3.8

Add $- \frac{y - x y'}{y^{2}}$ and $0$ .

$- 3 x^{- 2} - \frac{y - x y'}{y^{2}}$

Step 2.4

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

$- 3 \frac{1}{x^{2}} - \frac{y - x y'}{y^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 3$ and $\frac{1}{x^{2}}$ .

$\frac{- 3}{x^{2}} - \frac{y - x y'}{y^{2}}$

Step 2.5.1.2

Move the negative in front of the fraction.

$- \frac{3}{x^{2}} - \frac{y - x y'}{y^{2}}$

Step 2.5.1.3

To write $- \frac{3}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

$- \frac{3}{x^{2}} \cdot \frac{y^{2}}{y^{2}} - \frac{y - x y'}{y^{2}}$

Step 2.5.1.4

To write $- \frac{y - x y'}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$- \frac{3}{x^{2}} \cdot \frac{y^{2}}{y^{2}} - \frac{y - x y'}{y^{2}} \cdot \frac{x^{2}}{x^{2}}$

Step 2.5.1.5

Write each expression with a common denominator of $x^{2} y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1.5.1

Multiply $\frac{3}{x^{2}}$ by $\frac{y^{2}}{y^{2}}$ .

$- \frac{3 y^{2}}{x^{2} y^{2}} - \frac{y - x y'}{y^{2}} \cdot \frac{x^{2}}{x^{2}}$

Step 2.5.1.5.2

Multiply $\frac{y - x y'}{y^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

$- \frac{3 y^{2}}{x^{2} y^{2}} - \frac{(y - x y') x^{2}}{y^{2} x^{2}}$

Step 2.5.1.5.3

Reorder the factors of $y^{2} x^{2}$ .

$- \frac{3 y^{2}}{x^{2} y^{2}} - \frac{(y - x y') x^{2}}{x^{2} y^{2}}$

Step 2.5.1.6

Combine the numerators over the common denominator.

$\frac{- 3 y^{2} - (y - x y') x^{2}}{x^{2} y^{2}}$

Step 2.5.2

Reorder terms.

$\frac{- 3 y^{2} - x^{2} (y - x y')}{x^{2} y^{2}}$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0$

Step 4

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

$\frac{- 3 y^{2} - x^{2} (y - x y')}{x^{2} y^{2}} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$- 3 y^{2} - x^{2} (y - x y') = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the distributive property.

$- 3 y^{2} - x^{2} y - x^{2} (- x y') = 0$

Step 5.2.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.2.1.2.1

Move $x$ .

$- 3 y^{2} - x^{2} y - (x \cdot x^{2}) (- 1 y') = 0$

Step 5.2.1.2.2

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

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Step 5.2.1.2.2.1

Raise $x$ to the power of $1$ .

$- 3 y^{2} - x^{2} y - (x^{1} x^{2}) (- 1 y') = 0$

Step 5.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 3 y^{2} - x^{2} y - x^{1 + 2} (- 1 y') = 0$

Step 5.2.1.2.3

Add $1$ and $2$ .

$- 3 y^{2} - x^{2} y - x^{3} (- 1 y') = 0$

Step 5.2.1.3

Simplify each term.

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Step 5.2.1.3.1

Rewrite using the commutative property of multiplication.

$- 3 y^{2} - x^{2} y - 1 \cdot - 1 x^{3} y' = 0$

Step 5.2.1.3.2

Multiply $- 1$ by $- 1$ .

$- 3 y^{2} - x^{2} y + 1 x^{3} y' = 0$

Step 5.2.1.3.3

Multiply $x^{3}$ by $1$ .

$- 3 y^{2} - x^{2} y + x^{3} y' = 0$

Step 5.2.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.2.2.1

Add $3 y^{2}$ to both sides of the equation.

$- x^{2} y + x^{3} y' = 3 y^{2}$

Step 5.2.2.2

Add $x^{2} y$ to both sides of the equation.

$x^{3} y' = 3 y^{2} + x^{2} y$

Step 5.2.3

Divide each term in $x^{3} y' = 3 y^{2} + x^{2} y$ by $x^{3}$ and simplify.

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Step 5.2.3.1

Divide each term in $x^{3} y' = 3 y^{2} + x^{2} y$ by $x^{3}$ .

$\frac{x^{3} y'}{x^{3}} = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} y}{x^{3}}$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $x^{3}$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{x^{3} y'}{x^{3}} = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} y}{x^{3}}$

Step 5.2.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} y}{x^{3}}$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.2.3.3.1.1

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

$y' = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} (y)}{x^{3}}$

Step 5.2.3.3.1.2

Cancel the common factors.

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Step 5.2.3.3.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$y' = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} (y)}{x^{2} x}$

Step 5.2.3.3.1.2.2

Cancel the common factor.

$y' = \frac{3 y^{2}}{x^{3}} + \frac{x^{2} y}{x^{2} x}$

Step 5.2.3.3.1.2.3

Rewrite the expression.

$y' = \frac{3 y^{2}}{x^{3}} + \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = \frac{3 y^{2}}{x^{3}} + \frac{y}{x}$