Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $- 1$ by $3$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

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 $g (x) = y$ .

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

Step 2.3.3

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

$- 3 x^{- 2} - 3 \frac{y \cdot 0 - 1 \cdot 1 \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.4

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

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

Step 2.3.5

Multiply $y$ by $0$ .

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

Step 2.3.6

Multiply $- 1$ by $1$ .

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

Step 2.3.7

Subtract $y'$ from $0$ .

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

Step 2.3.8

Move the negative in front of the fraction.

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

Step 2.3.9

Multiply $- 1$ by $- 3$ .

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

Step 2.3.10

Combine $3$ and $\frac{y'}{y^{2}}$ .

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

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

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

Step 2.4.2.2

Move the negative in front of the fraction.

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

Step 2.4.3

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$2$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Add $\frac{3}{x^{2}}$ to both sides of the equation.

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

Step 5.2

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

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

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

Simplify the left side.

Tap for more steps...

Step 5.3.1.1

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

Tap for more steps...

Step 5.3.1.1.1

Cancel the common factor.

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

Step 5.3.1.1.2

Rewrite the expression.

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

Step 5.3.2

Simplify the right side.

Tap for more steps...

Step 5.3.2.1

Simplify $(2 + \frac{3}{x^{2}}) y^{2}$ .

Tap for more steps...

Step 5.3.2.1.1

Apply the distributive property.

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

Step 5.3.2.1.2

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

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

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Simplify each term.

Tap for more steps...

Step 5.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4.3.1.2

Combine.

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

Step 5.4.3.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.4.3.1.3.1

Cancel the common factor.

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

Step 5.4.3.1.3.2

Rewrite the expression.

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

Step 5.4.3.1.4

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

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

Step 6

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

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