Algebra Examples

$7 x y - \frac{y}{7} = \frac{2}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7 x y - \frac{y}{7}) = \frac{d}{d x} (\frac{2}{x})$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $7 x y - \frac{y}{7}$ with respect to $x$ is $\frac{d}{d x} [7 x y] + \frac{d}{d x} [- \frac{y}{7}]$ .

$\frac{d}{d x} [7 x y] + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.2

Evaluate $\frac{d}{d x} [7 x y]$ .

Tap for more steps...

Step 2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x y$ with respect to $x$ is $7 \frac{d}{d x} [x y]$ .

$7 \frac{d}{d x} [x y] + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.2.2

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

$7 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.2.3

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

$7 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.2.4

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

$7 (x y' + y \cdot 1) + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.2.5

Multiply $y$ by $1$ .

$7 (x y' + y) + \frac{d}{d x} [- \frac{y}{7}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{y}{7}]$ .

Tap for more steps...

Step 2.3.1

Since $- \frac{1}{7}$ is constant with respect to $x$ , the derivative of $- \frac{y}{7}$ with respect to $x$ is $- \frac{1}{7} \frac{d}{d x} [y]$ .

$7 (x y' + y) - \frac{1}{7} \frac{d}{d x} [y]$

Step 2.3.2

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

$7 (x y' + y) - \frac{1}{7} y'$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$7 (x y') + 7 y - \frac{1}{7} y'$

Step 2.4.2

Reorder terms.

$7 x y' - \frac{1}{7} y' + 7 y$

Step 2.4.3

Combine $y'$ and $\frac{1}{7}$ .

$7 x y' - \frac{y'}{7} + 7 y$

Step 2.4.4

To write $7 x y'$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$7 x y' \cdot \frac{7}{7} - \frac{y'}{7} + 7 y$

Step 2.4.5

Combine $7 x y'$ and $\frac{7}{7}$ .

$\frac{7 x y' \cdot 7}{7} - \frac{y'}{7} + 7 y$

Step 2.4.6

Combine the numerators over the common denominator.

$\frac{7 x y' \cdot 7 - y'}{7} + 7 y$

Step 2.4.7

Simplify the numerator.

Tap for more steps...

Step 2.4.7.1

Factor $y'$ out of $7 x y' \cdot 7 - y'$ .

Tap for more steps...

Step 2.4.7.1.1

Factor $y'$ out of $7 x y' \cdot 7$ .

$\frac{y' (7 x \cdot 7) - y'}{7} + 7 y$

Step 2.4.7.1.2

Factor $y'$ out of $- y'$ .

$\frac{y' (7 x \cdot 7) + y' \cdot - 1}{7} + 7 y$

Step 2.4.7.1.3

Factor $y'$ out of $y' (7 x \cdot 7) + y' \cdot - 1$ .

$\frac{y' (7 x \cdot 7 - 1)}{7} + 7 y$

Step 2.4.7.2

Multiply $7$ by $7$ .

$\frac{y' (49 x - 1)}{7} + 7 y$

Step 2.4.8

To write $7 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{y' (49 x - 1)}{7} + 7 y \cdot \frac{7}{7}$

Step 2.4.9

Combine $7 y$ and $\frac{7}{7}$ .

$\frac{y' (49 x - 1)}{7} + \frac{7 y \cdot 7}{7}$

Step 2.4.10

Combine the numerators over the common denominator.

$\frac{y' (49 x - 1) + 7 y \cdot 7}{7}$

Step 2.4.11

Simplify the numerator.

Tap for more steps...

Step 2.4.11.1

Apply the distributive property.

$\frac{y' (49 x) + y' \cdot - 1 + 7 y \cdot 7}{7}$

Step 2.4.11.2

Rewrite using the commutative property of multiplication.

$\frac{49 y' x + y' \cdot - 1 + 7 y \cdot 7}{7}$

Step 2.4.11.3

Move $- 1$ to the left of $y'$ .

$\frac{49 y' x - 1 \cdot y' + 7 y \cdot 7}{7}$

Step 2.4.11.4

Rewrite $- 1 y'$ as $- y'$ .

$\frac{49 y' x - y' + 7 y \cdot 7}{7}$

Step 2.4.11.5

Multiply $7$ by $7$ .

$\frac{49 y' x - y' + 49 y}{7}$

Step 2.4.12

Reorder factors in $\frac{49 y' x - y' + 49 y}{7}$ .

$\frac{49 x y' - y' + 49 y}{7}$

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 $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$2 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

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

Step 3.3

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

$2 (- x^{- 2})$

Step 3.4

Multiply $- 1$ by $2$ .

$- 2 x^{- 2}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

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

$- 2 \frac{1}{x^{2}}$

Step 3.5.2

Combine terms.

Tap for more steps...

Step 3.5.2.1

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

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

Step 3.5.2.2

Move the negative in front of the fraction.

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

Step 4

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

$\frac{49 x y' - y' + 49 y}{7} = - \frac{2}{x^{2}}$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Multiply both sides by $7$ .

$\frac{49 x y' - y' + 49 y}{7} \cdot 7 = - \frac{2}{x^{2}} \cdot 7$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

Simplify $\frac{49 x y' - y' + 49 y}{7} \cdot 7$ .

Tap for more steps...

Step 5.2.1.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{49 x y' - y' + 49 y}{7} \cdot 7 = - \frac{2}{x^{2}} \cdot 7$

Step 5.2.1.1.1.2

Rewrite the expression.

$49 x y' - y' + 49 y = - \frac{2}{x^{2}} \cdot 7$

Step 5.2.1.1.2

Move $x$ .

$49 y' x - y' + 49 y = - \frac{2}{x^{2}} \cdot 7$

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Simplify $- \frac{2}{x^{2}} \cdot 7$ .

Tap for more steps...

Step 5.2.2.1.1

Multiply $- \frac{2}{x^{2}} \cdot 7$ .

Tap for more steps...

Step 5.2.2.1.1.1

Multiply $7$ by $- 1$ .

$49 y' x - y' + 49 y = - 7 \frac{2}{x^{2}}$

Step 5.2.2.1.1.2

Combine $- 7$ and $\frac{2}{x^{2}}$ .

$49 y' x - y' + 49 y = \frac{- 7 \cdot 2}{x^{2}}$

Step 5.2.2.1.1.3

Multiply $- 7$ by $2$ .

$49 y' x - y' + 49 y = \frac{- 14}{x^{2}}$

Step 5.2.2.1.2

Move the negative in front of the fraction.

$49 y' x - y' + 49 y = - \frac{14}{x^{2}}$

Step 5.3

Solve for $y'$ .

Tap for more steps...

Step 5.3.1

Subtract $49 y$ from both sides of the equation.

$49 y' x - y' = - \frac{14}{x^{2}} - 49 y$

Step 5.3.2

Factor $y'$ out of $49 y' x - y'$ .

Tap for more steps...

Step 5.3.2.1

Factor $y'$ out of $49 y' x$ .

$y' (49 x) - y' = - \frac{14}{x^{2}} - 49 y$

Step 5.3.2.2

Factor $y'$ out of $- y'$ .

$y' (49 x) + y' \cdot - 1 = - \frac{14}{x^{2}} - 49 y$

Step 5.3.2.3

Factor $y'$ out of $y' (49 x) + y' \cdot - 1$ .

$y' (49 x - 1) = - \frac{14}{x^{2}} - 49 y$

Step 5.3.3

Divide each term in $y' (49 x - 1) = - \frac{14}{x^{2}} - 49 y$ by $49 x - 1$ and simplify.

Tap for more steps...

Step 5.3.3.1

Divide each term in $y' (49 x - 1) = - \frac{14}{x^{2}} - 49 y$ by $49 x - 1$ .

$\frac{y' (49 x - 1)}{49 x - 1} = \frac{- \frac{14}{x^{2}}}{49 x - 1} + \frac{- 49 y}{49 x - 1}$

Step 5.3.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.3.2.1

Cancel the common factor of $49 x - 1$ .

Tap for more steps...

Step 5.3.3.2.1.1

Cancel the common factor.

$\frac{y' (49 x - 1)}{49 x - 1} = \frac{- \frac{14}{x^{2}}}{49 x - 1} + \frac{- 49 y}{49 x - 1}$

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{14}{x^{2}}}{49 x - 1} + \frac{- 49 y}{49 x - 1}$

Step 5.3.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.3.1

Simplify each term.

Tap for more steps...

Step 5.3.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{14}{x^{2}} \cdot \frac{1}{49 x - 1} + \frac{- 49 y}{49 x - 1}$

Step 5.3.3.3.1.2

Multiply $\frac{1}{49 x - 1}$ by $\frac{14}{x^{2}}$ .

$y' = - \frac{14}{(49 x - 1) x^{2}} + \frac{- 49 y}{49 x - 1}$

Step 5.3.3.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{14}{(49 x - 1) x^{2}} - \frac{49 y}{49 x - 1}$

Step 5.3.3.3.2

To write $- \frac{49 y}{49 x - 1}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$y' = - \frac{14}{(49 x - 1) x^{2}} - \frac{49 y}{49 x - 1} \cdot \frac{x^{2}}{x^{2}}$

Step 5.3.3.3.3

Multiply $\frac{49 y}{49 x - 1}$ by $\frac{x^{2}}{x^{2}}$ .

$y' = - \frac{14}{(49 x - 1) x^{2}} - \frac{49 y x^{2}}{(49 x - 1) x^{2}}$

Step 5.3.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 14 - 49 y x^{2}}{(49 x - 1) x^{2}}$

Step 5.3.3.3.5

Factor $7$ out of $- 14 - 49 y x^{2}$ .

Tap for more steps...

Step 5.3.3.3.5.1

Factor $7$ out of $- 14$ .

$y' = \frac{7 (- 2) - 49 y x^{2}}{(49 x - 1) x^{2}}$

Step 5.3.3.3.5.2

Factor $7$ out of $- 49 y x^{2}$ .

$y' = \frac{7 (- 2) + 7 (- 7 y x^{2})}{(49 x - 1) x^{2}}$

Step 5.3.3.3.5.3

Factor $7$ out of $7 (- 2) + 7 (- 7 y x^{2})$ .

$y' = \frac{7 (- 2 - 7 y x^{2})}{(49 x - 1) x^{2}}$

Step 5.3.3.3.6

Rewrite $- 2$ as $- 1 (2)$ .

$y' = \frac{7 (- 1 (2) - 7 y x^{2})}{(49 x - 1) x^{2}}$

Step 5.3.3.3.7

Factor $- 1$ out of $- 7 y x^{2}$ .

$y' = \frac{7 (- 1 (2) - (7 y x^{2}))}{(49 x - 1) x^{2}}$

Step 5.3.3.3.8

Factor $- 1$ out of $- 1 (2) - (7 y x^{2})$ .

$y' = \frac{7 (- 1 (2 + 7 y x^{2}))}{(49 x - 1) x^{2}}$

Step 5.3.3.3.9

Simplify the expression.

Tap for more steps...

Step 5.3.3.3.9.1

Move the negative in front of the fraction.

$y' = - \frac{7 (2 + 7 y x^{2})}{(49 x - 1) x^{2}}$

Step 5.3.3.3.9.2

Reorder factors in $- \frac{7 (2 + 7 y x^{2})}{(49 x - 1) x^{2}}$ .

$y' = - \frac{7 (2 + 7 y x^{2})}{x^{2} (49 x - 1)}$

Step 6

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

$\frac{d y}{d x} = - \frac{7 (2 + 7 y x^{2})}{x^{2} (49 x - 1)}$