Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{2 x + y}{x - 5 y}) = \frac{d}{d x} (1)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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 $f (x) = 2 x + y$ and $g (x) = x - 5 y$ .

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

By the Sum Rule, the derivative of $2 x + y$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [y]$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify.

Tap for more steps...

Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Simplify the numerator.

Tap for more steps...

Step 2.8.2.1

Simplify each term.

Tap for more steps...

Step 2.8.2.1.1

Expand $(x - 5 y) (2 + y')$ using the FOIL Method.

Tap for more steps...

Step 2.8.2.1.1.1

Apply the distributive property.

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

Step 2.8.2.1.1.2

Apply the distributive property.

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

Step 2.8.2.1.1.3

Apply the distributive property.

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

Step 2.8.2.1.2

Simplify each term.

Tap for more steps...

Step 2.8.2.1.2.1

Move $2$ to the left of $x$ .

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

Step 2.8.2.1.2.2

Multiply $2$ by $- 5$ .

$\frac{2 x + x y' - 10 y - 5 y y' + (- (2 x) - y) (1 - 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.3

Multiply $2$ by $- 1$ .

$\frac{2 x + x y' - 10 y - 5 y y' + (- 2 x - y) (1 - 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.4

Expand $(- 2 x - y) (1 - 5 y')$ using the FOIL Method.

Tap for more steps...

Step 2.8.2.1.4.1

Apply the distributive property.

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x (1 - 5 y') - y (1 - 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.4.2

Apply the distributive property.

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x \cdot 1 - 2 x (- 5 y') - y (1 - 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.4.3

Apply the distributive property.

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x \cdot 1 - 2 x (- 5 y') - y \cdot 1 - y (- 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.5

Simplify each term.

Tap for more steps...

Step 2.8.2.1.5.1

Multiply $- 2$ by $1$ .

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x - 2 x (- 5 y') - y \cdot 1 - y (- 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.5.2

Multiply $- 5$ by $- 2$ .

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x + 10 x y' - y \cdot 1 - y (- 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.5.3

Multiply $- 1$ by $1$ .

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x + 10 x y' - y - y (- 5 y')}{{(x - 5 y)}^{2}}$

Step 2.8.2.1.5.4

Multiply $- 5$ by $- 1$ .

$\frac{2 x + x y' - 10 y - 5 y y' - 2 x + 10 x y' - y + 5 y y'}{{(x - 5 y)}^{2}}$

Step 2.8.2.2

Combine the opposite terms in $2 x + x y' - 10 y - 5 y y' - 2 x + 10 x y' - y + 5 y y'$ .

Tap for more steps...

Step 2.8.2.2.1

Subtract $2 x$ from $2 x$ .

$\frac{x y' - 10 y - 5 y y' + 0 + 10 x y' - y + 5 y y'}{{(x - 5 y)}^{2}}$

Step 2.8.2.2.2

Add $x y' - 10 y - 5 y y'$ and $0$ .

$\frac{x y' - 10 y - 5 y y' + 10 x y' - y + 5 y y'}{{(x - 5 y)}^{2}}$

Step 2.8.2.2.3

Add $- 5 y y'$ and $5 y y'$ .

$\frac{x y' - 10 y + 10 x y' - y + 0}{{(x - 5 y)}^{2}}$

Step 2.8.2.2.4

Add $x y' - 10 y + 10 x y' - y$ and $0$ .

$\frac{x y' - 10 y + 10 x y' - y}{{(x - 5 y)}^{2}}$

Step 2.8.2.3

Add $x y'$ and $10 x y'$ .

$\frac{11 x y' - 10 y - y}{{(x - 5 y)}^{2}}$

Step 2.8.2.4

Subtract $y$ from $- 10 y$ .

$\frac{11 x y' - 11 y}{{(x - 5 y)}^{2}}$

Step 2.8.3

Factor $11$ out of $11 x y' - 11 y$ .

Tap for more steps...

Step 2.8.3.1

Factor $11$ out of $11 x y'$ .

$\frac{11 (x y') - 11 y}{{(x - 5 y)}^{2}}$

Step 2.8.3.2

Factor $11$ out of $- 11 y$ .

$\frac{11 (x y') + 11 (- y)}{{(x - 5 y)}^{2}}$

Step 2.8.3.3

Factor $11$ out of $11 (x y') + 11 (- y)$ .

$\frac{11 (x y' - y)}{{(x - 5 y)}^{2}}$

Step 3

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

$0$

Step 4

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

$\frac{11 (x y' - y)}{{(x - 5 y)}^{2}} = 0$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Set the numerator equal to zero.

$11 (x y' - y) = 0$

Step 5.2

Solve the equation for $y'$ .

Tap for more steps...

Step 5.2.1

Divide each term in $11 (x y' - y) = 0$ by $11$ and simplify.

Tap for more steps...

Step 5.2.1.1

Divide each term in $11 (x y' - y) = 0$ by $11$ .

$\frac{11 (x y' - y)}{11} = \frac{0}{11}$

Step 5.2.1.2

Simplify the left side.

Tap for more steps...

Step 5.2.1.2.1

Cancel the common factor of $11$ .

Tap for more steps...

Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{11 (x y' - y)}{11} = \frac{0}{11}$

Step 5.2.1.2.1.2

Divide $x y' - y$ by $1$ .

$x y' - y = \frac{0}{11}$

Step 5.2.1.3

Simplify the right side.

Tap for more steps...

Step 5.2.1.3.1

Divide $0$ by $11$ .

$x y' - y = 0$

Step 5.2.2

Add $y$ to both sides of the equation.

$x y' = y$

Step 5.2.3

Divide each term in $x y' = y$ by $x$ and simplify.

Tap for more steps...

Step 5.2.3.1

Divide each term in $x y' = y$ by $x$ .

$\frac{x y'}{x} = \frac{y}{x}$

Step 5.2.3.2

Simplify the left side.

Tap for more steps...

Step 5.2.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{y}{x}$

Step 5.2.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = \frac{y}{x}$