Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $- 3 - 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 3] + \frac{d}{d x} [- 4 x^{3}]$ .

$\frac{d}{d x} [- 3] + \frac{d}{d x} [- 4 x^{3}]$

Step 2.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 4 x^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$0 - 4 \frac{d}{d x} [x^{3}]$

Step 2.2.2

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

$0 - 4 (3 x^{2})$

Step 2.2.3

Multiply $3$ by $- 4$ .

$0 - 12 x^{2}$

Step 2.3

Subtract $12 x^{2}$ from $0$ .

$- 12 x^{2}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$2 y' + \frac{d}{d x} [x y]$

Step 3.3

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

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Step 3.3.1

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 $g (x) = y$ .

$2 y' + x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 3.3.2

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

$2 y' + x y' + y \frac{d}{d x} [x]$

Step 3.3.3

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

$2 y' + x y' + y \cdot 1$

Step 3.3.4

Multiply $y$ by $1$ .

$2 y' + x y' + y$

Step 3.4

Reorder terms.

$x y' + y + 2 y'$

Step 4

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

$- 12 x^{2} = x y' + y + 2 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x y' + y + 2 y' = - 12 x^{2}$ .

$x y' + y + 2 y' = - 12 x^{2}$

Step 5.2

Subtract $y$ from both sides of the equation.

$x y' + 2 y' = - 12 x^{2} - y$

Step 5.3

Factor $y'$ out of $x y' + 2 y'$ .

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Step 5.3.1

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

$y' x + 2 y' = - 12 x^{2} - y$

Step 5.3.2

Factor $y'$ out of $2 y'$ .

$y' x + y' \cdot 2 = - 12 x^{2} - y$

Step 5.3.3

Factor $y'$ out of $y' x + y' \cdot 2$ .

$y' (x + 2) = - 12 x^{2} - y$

Step 5.4

Divide each term in $y' (x + 2) = - 12 x^{2} - y$ by $x + 2$ and simplify.

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Step 5.4.1

Divide each term in $y' (x + 2) = - 12 x^{2} - y$ by $x + 2$ .

$\frac{y' (x + 2)}{x + 2} = \frac{- 12 x^{2}}{x + 2} + \frac{- y}{x + 2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x + 2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (x + 2)}{x + 2} = \frac{- 12 x^{2}}{x + 2} + \frac{- y}{x + 2}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 12 x^{2}}{x + 2} + \frac{- y}{x + 2}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{12 x^{2}}{x + 2} + \frac{- y}{x + 2}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{12 x^{2}}{x + 2} - \frac{y}{x + 2}$

Step 5.4.3.2

Simplify terms.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 12 x^{2} - y}{x + 2}$

Step 5.4.3.2.2

Factor $- 1$ out of $- 12 x^{2}$ .

$y' = \frac{- (12 x^{2}) - y}{x + 2}$

Step 5.4.3.2.3

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

$y' = \frac{- (12 x^{2}) - (y)}{x + 2}$

Step 5.4.3.2.4

Factor $- 1$ out of $- (12 x^{2}) - (y)$ .

$y' = \frac{- (12 x^{2} + y)}{x + 2}$

Step 5.4.3.2.5

Simplify the expression.

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Step 5.4.3.2.5.1

Rewrite $- (12 x^{2} + y)$ as $- 1 (12 x^{2} + y)$ .

$y' = \frac{- 1 (12 x^{2} + y)}{x + 2}$

Step 5.4.3.2.5.2

Move the negative in front of the fraction.

$y' = - \frac{12 x^{2} + y}{x + 2}$

Step 6

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

$\frac{d y}{d x} = - \frac{12 x^{2} + y}{x + 2}$