Calculus Examples

$x^{3} - x^{2} - x y = 4$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} - x^{2} - x y) = \frac{d}{d x} (4)$

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 $x^{3} - x^{2} - x y$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- x y]$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.2.3

Multiply $2$ by $- 1$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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 $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

$3 x^{2} - 2 x - (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Remove unnecessary parentheses.

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

Step 2.4.3

Reorder terms.

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

Step 3

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

$0$

Step 4

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

$3 x^{2} - x y' - 2 x - y = 0$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 5.1.2

Add $2 x$ to both sides of the equation.

$- x y' - y = - 3 x^{2} + 2 x$

Step 5.1.3

Add $y$ to both sides of the equation.

$- x y' = - 3 x^{2} + 2 x + y$

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 5.2.3.1.1.1

Factor $x$ out of $- 3 x^{2}$ .

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

Step 5.2.3.1.1.2

Move the negative one from the denominator of $\frac{- 3 x}{- 1}$ .

$y' = - 1 \cdot (- 3 x) + \frac{2 x}{- x} + \frac{y}{- x}$

Step 5.2.3.1.2

Rewrite $- 1 \cdot (- 3 x)$ as $- (- 3 x)$ .

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

Step 5.2.3.1.3

Multiply $- 3$ by $- 1$ .

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

Step 5.2.3.1.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.3.1.4.1

Cancel the common factor.

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

Step 5.2.3.1.4.2

Rewrite the expression.

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

Step 5.2.3.1.4.3

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

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

Step 5.2.3.1.5

Multiply $- 1$ by $2$ .

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

Step 5.2.3.1.6

Move the negative in front of the fraction.

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

Step 6

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

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