Calculus Examples

$x^{2} (y - 6) = 12 y + 3$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} (y - 6)) = \frac{d}{d x} (12 y + 3)$

Step 2

Differentiate the left side of the equation.

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Step 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 $f (x) = x^{2}$ and $g (x) = y - 6$ .

$x^{2} \frac{d}{d x} [y - 6] + (y - 6) \frac{d}{d x} [x^{2}]$

Step 2.2

By the Sum Rule, the derivative of $y - 6$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [- 6]$ .

$x^{2} (\frac{d}{d x} [y] + \frac{d}{d x} [- 6]) + (y - 6) \frac{d}{d x} [x^{2}]$

Step 2.3

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

$x^{2} (y' + \frac{d}{d x} [- 6]) + (y - 6) \frac{d}{d x} [x^{2}]$

Step 2.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$x^{2} (y' + 0) + (y - 6) \frac{d}{d x} [x^{2}]$

Step 2.5

Add $y'$ and $0$ .

$x^{2} y' + (y - 6) \frac{d}{d x} [x^{2}]$

Step 2.6

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

$x^{2} y' + (y - 6) (2 x)$

Step 2.7

Move $2$ to the left of $y - 6$ .

$x^{2} y' + 2 \cdot (y - 6) x$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$x^{2} y' + (2 y + 2 \cdot - 6) x$

Step 2.8.2

Apply the distributive property.

$x^{2} y' + 2 y x + 2 \cdot - 6 x$

Step 2.8.3

Multiply $2$ by $- 6$ .

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

Step 2.8.4

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $12 y + 3$ with respect to $x$ is $\frac{d}{d x} [12 y] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [12 y] + \frac{d}{d x} [3]$

Step 3.2

Evaluate $\frac{d}{d x} [12 y]$ .

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Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 y$ with respect to $x$ is $12 \frac{d}{d x} [y]$ .

$12 \frac{d}{d x} [y] + \frac{d}{d x} [3]$

Step 3.2.2

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

$12 y' + \frac{d}{d x} [3]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$12 y' + 0$

Step 3.3.2

Add $12 y'$ and $0$ .

$12 y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $12 y'$ from both sides of the equation.

$x^{2} y' + 2 x y - 12 x - 12 y' = 0$

Step 5.2

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

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Step 5.2.1

Subtract $2 x y$ from both sides of the equation.

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

Step 5.2.2

Add $12 x$ to both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

Factor $y'$ out of $x^{2} y'$ .

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

Step 5.3.2

Factor $y'$ out of $- 12 y'$ .

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

Step 5.3.3

Factor $y'$ out of $y' x^{2} + y' \cdot - 12$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{2} - 12$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

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

Step 5.4.3.2

Simplify the numerator.

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Step 5.4.3.2.1

Factor $2 x$ out of $- 2 x y + 12 x$ .

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Step 5.4.3.2.1.1

Factor $2 x$ out of $- 2 x y$ .

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

Step 5.4.3.2.1.2

Factor $2 x$ out of $12 x$ .

$y' = \frac{2 x (- 1 y) + 2 x (6)}{x^{2} - 12}$

Step 5.4.3.2.1.3

Factor $2 x$ out of $2 x (- 1 y) + 2 x (6)$ .

$y' = \frac{2 x (- 1 y + 6)}{x^{2} - 12}$

Step 5.4.3.2.2

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

$y' = \frac{2 x (- y + 6)}{x^{2} - 12}$

Step 5.4.3.3

Simplify with factoring out.

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Step 5.4.3.3.1

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

$y' = \frac{2 x (- (y) + 6)}{x^{2} - 12}$

Step 5.4.3.3.2

Rewrite $6$ as $- 1 (- 6)$ .

$y' = \frac{2 x (- (y) - 1 (- 6))}{x^{2} - 12}$

Step 5.4.3.3.3

Factor $- 1$ out of $- (y) - 1 (- 6)$ .

$y' = \frac{2 x (- (y - 6))}{x^{2} - 12}$

Step 5.4.3.3.4

Simplify the expression.

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Step 5.4.3.3.4.1

Rewrite $- (y - 6)$ as $- 1 (y - 6)$ .

$y' = \frac{2 x (- 1 (y - 6))}{x^{2} - 12}$

Step 5.4.3.3.4.2

Move the negative in front of the fraction.

$y' = - \frac{(2 x) (y - 6)}{x^{2} - 12}$

Step 6

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

$\frac{d y}{d x} = - \frac{(2 x) (y - 6)}{x^{2} - 12}$