Calculus Examples

$3 x y - x^{2} - 4 = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x y - x^{2} - 4) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x y - x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [3 x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [3 x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x y]$ .

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Step 2.2.1

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

$3 \frac{d}{d x} [x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4]$

Step 2.2.2

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 $g (x) = y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$3 (x y' + y \cdot 1) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4]$

Step 2.2.5

Multiply $y$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$3 (x y' + y) - (2 x) + \frac{d}{d x} [- 4]$

Step 2.3.3

Multiply $2$ by $- 1$ .

$3 (x y' + y) - 2 x + \frac{d}{d x} [- 4]$

Step 2.4

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

$3 (x y' + y) - 2 x + 0$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$3 (x y') + 3 y - 2 x + 0$

Step 2.5.2

Add $3 x y' + 3 y - 2 x$ and $0$ .

$3 x y' + 3 y - 2 x$

Step 2.5.3

Reorder terms.

$3 x y' - 2 x + 3 y$

Step 3

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

$0$

Step 4

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

$3 x y' - 2 x + 3 y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

$3 x y' + 3 y = 2 x$

Step 5.1.2

Subtract $3 y$ from both sides of the equation.

$3 x y' = 2 x - 3 y$

Step 5.2

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

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Step 5.2.1

Divide each term in $3 x y' = 2 x - 3 y$ by $3 x$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $x$ .

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Step 5.2.3.1.1.1

Cancel the common factor.

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

Step 5.2.3.1.1.2

Rewrite the expression.

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

Step 5.2.3.1.2

Cancel the common factor of $- 3$ and $3$ .

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Step 5.2.3.1.2.1

Factor $3$ out of $- 3 y$ .

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

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

Factor $3$ out of $3 x$ .

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

Step 5.2.3.1.2.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.2.3

Rewrite the expression.

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

Step 5.2.3.1.3

Move the negative in front of the fraction.

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

Step 6

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

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