Calculus Examples

$2 x^{2} y - \frac{3}{y} = 0$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.3

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

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Step 2.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{y}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{y}]$ .

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

Step 2.3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = y$ .

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

Step 2.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (x^{2} y' + 2 y x) - 3 \frac{y \cdot 0 - 1 \cdot 1 \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.4

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

$2 (x^{2} y' + 2 y x) - 3 \frac{y \cdot 0 - 1 \cdot 1 y'}{y^{2}}$

Step 2.3.5

Multiply $y$ by $0$ .

$2 (x^{2} y' + 2 y x) - 3 \frac{0 - 1 \cdot 1 y'}{y^{2}}$

Step 2.3.6

Multiply $- 1$ by $1$ .

$2 (x^{2} y' + 2 y x) - 3 \frac{0 - y'}{y^{2}}$

Step 2.3.7

Subtract $y'$ from $0$ .

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

Step 2.3.8

Move the negative in front of the fraction.

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

Step 2.3.9

Multiply $- 1$ by $- 3$ .

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

Step 2.3.10

Combine $3$ and $\frac{y'}{y^{2}}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $2$ by $2$ .

$2 x^{2} y' + 4 y x + \frac{3 y'}{y^{2}}$

Step 2.4.3

Reorder terms.

$2 x^{2} y' + 4 x y + \frac{3 y'}{y^{2}}$

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.

$2 x^{2} y' + 4 x y + \frac{3 y'}{y^{2}} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Find the LCD of the terms in the equation.

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Step 5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 1, y^{2}, 1$

Step 5.1.2

The LCM of one and any expression is the expression.

$y^{2}$

Step 5.2

Multiply each term in $2 x^{2} y' + 4 x y + \frac{3 y'}{y^{2}} = 0$ by $y^{2}$ to eliminate the fractions.

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Step 5.2.1

Multiply each term in $2 x^{2} y' + 4 x y + \frac{3 y'}{y^{2}} = 0$ by $y^{2}$ .

$2 x^{2} y' y^{2} + 4 x y \cdot y^{2} + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 5.2.2.1.1.1

Move $y^{2}$ .

$2 x^{2} y' y^{2} + 4 x (y^{2} y) + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2.1.1.2

Multiply $y^{2}$ by $y$ .

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Step 5.2.2.1.1.2.1

Raise $y$ to the power of $1$ .

$2 x^{2} y' y^{2} + 4 x (y^{2} y^{1}) + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2} y' y^{2} + 4 x y^{2 + 1} + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2.1.1.3

Add $2$ and $1$ .

$2 x^{2} y' y^{2} + 4 x y^{3} + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2.1.2

Cancel the common factor of $y^{2}$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

$2 x^{2} y' y^{2} + 4 x y^{3} + \frac{3 y'}{y^{2}} y^{2} = 0 y^{2}$

Step 5.2.2.1.2.2

Rewrite the expression.

$2 x^{2} y' y^{2} + 4 x y^{3} + 3 y' = 0 y^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply $0$ by $y^{2}$ .

$2 x^{2} y' y^{2} + 4 x y^{3} + 3 y' = 0$

Step 5.3

Solve the equation.

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Step 5.3.1

Subtract $4 x y^{3}$ from both sides of the equation.

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

Step 5.3.2

Factor $y'$ out of $2 x^{2} y' y^{2} + 3 y'$ .

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Step 5.3.2.1

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

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

Step 5.3.2.2

Factor $y'$ out of $3 y'$ .

$y' (2 x^{2} y^{2}) + y' \cdot 3 = - 4 x y^{3}$

Step 5.3.2.3

Factor $y'$ out of $y' (2 x^{2} y^{2}) + y' \cdot 3$ .

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

Step 5.3.3

Divide each term in $y' (2 x^{2} y^{2} + 3) = - 4 x y^{3}$ by $2 x^{2} y^{2} + 3$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (2 x^{2} y^{2} + 3) = - 4 x y^{3}$ by $2 x^{2} y^{2} + 3$ .

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $2 x^{2} y^{2} + 3$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3.3

Simplify the right side.

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Step 5.3.3.3.1

Move the negative in front of the fraction.

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

Step 6

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

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