Calculus Examples

$4 + x y - 4 x = 5 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 + x y - 4 x) = \frac{d}{d x} (5 y)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $4 + x y - 4 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 4 x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 4 x]$

Step 2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x y] + \frac{d}{d x} [- 4 x]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$0 + x \frac{d}{d x} [y] + y \frac{d}{d x} [x] + \frac{d}{d x} [- 4 x]$

Step 2.2.2

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

$0 + x y' + y \frac{d}{d x} [x] + \frac{d}{d x} [- 4 x]$

Step 2.2.3

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

$0 + x y' + y \cdot 1 + \frac{d}{d x} [- 4 x]$

Step 2.2.4

Multiply $y$ by $1$ .

$0 + x y' + y + \frac{d}{d x} [- 4 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 4 x]$ .

Tap for more steps...

Step 2.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$0 + x y' + y - 4 \frac{d}{d x} [x]$

Step 2.3.2

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

$0 + x y' + y - 4 \cdot 1$

Step 2.3.3

Multiply $- 4$ by $1$ .

$0 + x y' + y - 4$

Step 2.4

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

$x y' + y - 4$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$5 y'$

Step 4

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

$x y' + y - 4 = 5 y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Subtract $5 y'$ from both sides of the equation.

$x y' + y - 4 - 5 y' = 0$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $y$ from both sides of the equation.

$x y' - 4 - 5 y' = - y$

Step 5.2.2

Add $4$ to both sides of the equation.

$x y' - 5 y' = - y + 4$

Step 5.3

Factor $y'$ out of $x y' - 5 y'$ .

Tap for more steps...

Step 5.3.1

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

$y' x - 5 y' = - y + 4$

Step 5.3.2

Factor $y'$ out of $- 5 y'$ .

$y' x + y' \cdot - 5 = - y + 4$

Step 5.3.3

Factor $y'$ out of $y' x + y' \cdot - 5$ .

$y' (x - 5) = - y + 4$

Step 5.4

Divide each term in $y' (x - 5) = - y + 4$ by $x - 5$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $y' (x - 5) = - y + 4$ by $x - 5$ .

$\frac{y' (x - 5)}{x - 5} = \frac{- y}{x - 5} + \frac{4}{x - 5}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $x - 5$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (x - 5)}{x - 5} = \frac{- y}{x - 5} + \frac{4}{x - 5}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- y}{x - 5} + \frac{4}{x - 5}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{- y + 4}{x - 5}$

Step 5.4.3.2

Factor $- 1$ out of $- y$ .

$y' = \frac{- (y) + 4}{x - 5}$

Step 5.4.3.3

Rewrite $4$ as $- 1 (- 4)$ .

$y' = \frac{- (y) - 1 (- 4)}{x - 5}$

Step 5.4.3.4

Factor $- 1$ out of $- (y) - 1 (- 4)$ .

$y' = \frac{- (y - 4)}{x - 5}$

Step 5.4.3.5

Simplify the expression.

Tap for more steps...

Step 5.4.3.5.1

Rewrite $- (y - 4)$ as $- 1 (y - 4)$ .

$y' = \frac{- 1 (y - 4)}{x - 5}$

Step 5.4.3.5.2

Move the negative in front of the fraction.

$y' = - \frac{y - 4}{x - 5}$

Step 6

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

$\frac{d y}{d x} = - \frac{y - 4}{x - 5}$