Calculus Examples

$x^{2} y^{2} - 2 x y + x^{2} = 1$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $x$ by $1$ .

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

Step 2.4

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

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Step 2.4.1

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

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Step 2.4.1.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

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

Step 2.4.1.2

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

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

Step 2.4.1.3

Replace all occurrences of $u_{2}$ with $x$ .

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

Step 2.4.2

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

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Remove unnecessary parentheses.

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

Step 2.5.3

Reorder terms.

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Factor $2 x'$ out of $2 x' (y^{2} x) + 2 x' (- y)$ .

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

Step 5.2.5

Factor $2 x'$ out of $2 x' (y^{2} x - y) + 2 x' x$ .

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $x'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 5.3.3.1.1.1

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

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.3

Cancel the common factor of $2$ .

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Step 5.3.3.1.3.1

Cancel the common factor.

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

Step 5.3.3.1.3.2

Rewrite the expression.

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

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2.2

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

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Step 5.3.3.2.2.1

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

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

Step 5.3.3.2.2.2

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

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

Step 5.3.3.2.2.3

Factor $x$ out of $x^{1}$ .

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

Step 5.3.3.2.2.4

Factor $x$ out of $x (- x y) + x \cdot 1$ .

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

Step 5.3.3.2.3

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

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

Step 5.3.3.2.4

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

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

Step 5.3.3.2.5

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

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

Step 5.3.3.2.6

Simplify the expression.

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Step 5.3.3.2.6.1

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

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

Step 5.3.3.2.6.2

Move the negative in front of the fraction.

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

Step 6

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

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