Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Move $2$ to the left of $x^{2}$ .

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

Step 2.2.6

Move $2$ to the left of $y^{2}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 2$ by $1$ .

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

Step 3

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

$0$

Step 4

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

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

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

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

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

Step 5.1.2

Add $2$ to both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

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

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Rewrite the expression.

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

Step 5.2.2.3

Cancel the common factor of $y$ .

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Step 5.2.2.3.1

Cancel the common factor.

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

Step 5.2.2.3.2

Divide $y'$ by $1$ .

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

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 $- 2$ and $2$ .

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Step 5.2.3.1.1.1

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

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

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

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

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

Step 5.2.3.1.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.2

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

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Step 5.2.3.1.2.1

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

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

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

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

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

Step 5.2.3.1.2.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.2.3

Rewrite the expression.

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

Step 5.2.3.1.3

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

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Step 5.2.3.1.3.1

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

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

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

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

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

Step 5.2.3.1.3.2.2

Cancel the common factor.

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

Step 5.2.3.1.3.2.3

Rewrite the expression.

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

Step 5.2.3.1.4

Move the negative in front of the fraction.

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

Step 5.2.3.1.5

Cancel the common factor of $2$ .

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Step 5.2.3.1.5.1

Cancel the common factor.

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

Step 5.2.3.1.5.2

Rewrite the expression.

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

Step 6

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

$\frac{d y}{d x} = - \frac{y}{x} + \frac{1}{x^{2} y}$