Calculus Examples

$\frac{5}{x} + \frac{5}{y} = 3$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{5}{x} + \frac{5}{y}) = \frac{d}{d x} (3)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $\frac{5}{x} + \frac{5}{y}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{x}] + \frac{d}{d x} [\frac{5}{y}]$ .

$\frac{d}{d x} [\frac{5}{x}] + \frac{d}{d x} [\frac{5}{y}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{5}{x}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x}]$ .

$5 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{5}{y}]$

Step 2.2.2

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

$5 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [\frac{5}{y}]$

Step 2.2.3

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

$5 (- x^{- 2}) + \frac{d}{d x} [\frac{5}{y}]$

Step 2.2.4

Multiply $- 1$ by $5$ .

$- 5 x^{- 2} + \frac{d}{d x} [\frac{5}{y}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{5}{y}]$ .

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Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{y}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{y}]$ .

$- 5 x^{- 2} + 5 \frac{d}{d x} [\frac{1}{y}]$

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

$- 5 x^{- 2} + 5 \frac{y \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.3

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

$- 5 x^{- 2} + 5 \frac{y \cdot 0 - 1 \cdot 1 \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.4

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

$- 5 x^{- 2} + 5 \frac{y \cdot 0 - 1 \cdot 1 y'}{y^{2}}$

Step 2.3.5

Multiply $y$ by $0$ .

$- 5 x^{- 2} + 5 \frac{0 - 1 \cdot 1 y'}{y^{2}}$

Step 2.3.6

Multiply $- 1$ by $1$ .

$- 5 x^{- 2} + 5 \frac{0 - y'}{y^{2}}$

Step 2.3.7

Subtract $y'$ from $0$ .

$- 5 x^{- 2} + 5 \frac{- y'}{y^{2}}$

Step 2.3.8

Move the negative in front of the fraction.

$- 5 x^{- 2} + 5 (- \frac{y'}{y^{2}})$

Step 2.3.9

Multiply $- 1$ by $5$ .

$- 5 x^{- 2} - 5 \frac{y'}{y^{2}}$

Step 2.3.10

Combine $- 5$ and $\frac{y'}{y^{2}}$ .

$- 5 x^{- 2} + \frac{- 5 y'}{y^{2}}$

Step 2.3.11

Move the negative in front of the fraction.

$- 5 x^{- 2} - \frac{5 y'}{y^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

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

$- 5 \frac{1}{x^{2}} - \frac{5 y'}{y^{2}}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 5$ and $\frac{1}{x^{2}}$ .

$\frac{- 5}{x^{2}} - \frac{5 y'}{y^{2}}$

Step 2.4.2.2

Move the negative in front of the fraction.

$- \frac{5}{x^{2}} - \frac{5 y'}{y^{2}}$

Step 3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- \frac{5}{x^{2}} - \frac{5 y'}{y^{2}} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Add $\frac{5}{x^{2}}$ to both sides of the equation.

$- \frac{5 y'}{y^{2}} = \frac{5}{x^{2}}$

Step 5.2

Divide each term in $- \frac{5 y'}{y^{2}} = \frac{5}{x^{2}}$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{5 y'}{y^{2}} = \frac{5}{x^{2}}$ by $- 1$ .

$\frac{- \frac{5 y'}{y^{2}}}{- 1} = \frac{\frac{5}{x^{2}}}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 y'}{y^{2}}}{1} = \frac{\frac{5}{x^{2}}}{- 1}$

Step 5.2.2.2

Divide $\frac{5 y'}{y^{2}}$ by $1$ .

$\frac{5 y'}{y^{2}} = \frac{\frac{5}{x^{2}}}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative one from the denominator of $\frac{\frac{5}{x^{2}}}{- 1}$ .

$\frac{5 y'}{y^{2}} = - 1 \cdot \frac{5}{x^{2}}$

Step 5.2.3.2

Rewrite $- 1 \cdot \frac{5}{x^{2}}$ as $- \frac{5}{x^{2}}$ .

$\frac{5 y'}{y^{2}} = - \frac{5}{x^{2}}$

Step 5.3

Multiply both sides by $y^{2}$ .

$\frac{5 y'}{y^{2}} y^{2} = - \frac{5}{x^{2}} y^{2}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{5 y'}{y^{2}} y^{2} = - \frac{5}{x^{2}} y^{2}$

Step 5.4.1.1.2

Rewrite the expression.

$5 y' = - \frac{5}{x^{2}} y^{2}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $- \frac{5}{x^{2}} y^{2}$ .

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Step 5.4.2.1.1

Combine $y^{2}$ and $\frac{5}{x^{2}}$ .

$5 y' = - \frac{y^{2} \cdot 5}{x^{2}}$

Step 5.4.2.1.2

Move $5$ to the left of $y^{2}$ .

$5 y' = - \frac{5 y^{2}}{x^{2}}$

Step 5.5

Divide each term in $5 y' = - \frac{5 y^{2}}{x^{2}}$ by $5$ and simplify.

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Step 5.5.1

Divide each term in $5 y' = - \frac{5 y^{2}}{x^{2}}$ by $5$ .

$\frac{5 y'}{5} = \frac{- \frac{5 y^{2}}{x^{2}}}{5}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{5 y'}{5} = \frac{- \frac{5 y^{2}}{x^{2}}}{5}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{5 y^{2}}{x^{2}}}{5}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{5 y^{2}}{x^{2}} \cdot \frac{1}{5}$

Step 5.5.3.2

Cancel the common factor of $5$ .

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Step 5.5.3.2.1

Move the leading negative in $- \frac{5 y^{2}}{x^{2}}$ into the numerator.

$y' = \frac{- 5 y^{2}}{x^{2}} \cdot \frac{1}{5}$

Step 5.5.3.2.2

Factor $5$ out of $- 5 y^{2}$ .

$y' = \frac{5 (- y^{2})}{x^{2}} \cdot \frac{1}{5}$

Step 5.5.3.2.3

Cancel the common factor.

$y' = \frac{5 (- y^{2})}{x^{2}} \cdot \frac{1}{5}$

Step 5.5.3.2.4

Rewrite the expression.

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

Step 5.5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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