Calculus Examples

$x^{2} y^{2} - 2 x y + x^{2} = 1$ , $(- 1, 0)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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 1.2.2.2.1

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

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

Step 1.2.2.2.2

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

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

Step 1.2.2.2.3

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

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

Step 1.2.2.3

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

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

Step 1.2.2.4

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

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

Step 1.2.2.5

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

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

Step 1.2.2.6

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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 $g (x) = y$ .

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

Step 1.2.3.3

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

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

Step 1.2.3.4

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

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

Step 1.2.3.5

Multiply $y$ by $1$ .

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

Step 1.2.4

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

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

Step 1.2.5

Simplify.

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Step 1.2.5.1

Apply the distributive property.

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

Step 1.2.5.2

Remove unnecessary parentheses.

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

Step 1.2.5.3

Reorder terms.

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

Step 1.3

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

$0$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.2.3

Factor $2 x y'$ out of $2 x y' (x y) + 2 x y' \cdot - 1$ .

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

Step 1.5.3

Divide each term in $2 x y' (x y - 1) = - 2 y^{2} x - 2 x + 2 y$ by $2 x (x y - 1)$ and simplify.

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Step 1.5.3.1

Divide each term in $2 x y' (x y - 1) = - 2 y^{2} x - 2 x + 2 y$ by $2 x (x y - 1)$ .

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $2$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of $x$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

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

Step 1.5.3.2.2.2

Rewrite the expression.

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

Step 1.5.3.2.3

Cancel the common factor of $x y - 1$ .

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Step 1.5.3.2.3.1

Cancel the common factor.

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

Step 1.5.3.2.3.2

Divide $y'$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Simplify each term.

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Step 1.5.3.3.1.1

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

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Step 1.5.3.3.1.1.1

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

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

Step 1.5.3.3.1.1.2

Cancel the common factors.

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Step 1.5.3.3.1.1.2.1

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

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

Step 1.5.3.3.1.1.2.2

Cancel the common factor.

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

Step 1.5.3.3.1.1.2.3

Rewrite the expression.

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

Step 1.5.3.3.1.2

Cancel the common factor of $x$ .

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Step 1.5.3.3.1.2.1

Cancel the common factor.

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

Step 1.5.3.3.1.2.2

Rewrite the expression.

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

Step 1.5.3.3.1.3

Move the negative in front of the fraction.

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

Step 1.5.3.3.1.4

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

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Step 1.5.3.3.1.4.1

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

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

Step 1.5.3.3.1.4.2

Cancel the common factors.

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Step 1.5.3.3.1.4.2.1

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

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

Step 1.5.3.3.1.4.2.2

Cancel the common factor.

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

Step 1.5.3.3.1.4.2.3

Rewrite the expression.

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

Step 1.5.3.3.1.5

Cancel the common factor of $x$ .

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Step 1.5.3.3.1.5.1

Cancel the common factor.

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

Step 1.5.3.3.1.5.2

Rewrite the expression.

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

Step 1.5.3.3.1.6

Move the negative in front of the fraction.

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

Step 1.5.3.3.1.7

Cancel the common factor of $2$ .

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Step 1.5.3.3.1.7.1

Cancel the common factor.

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

Step 1.5.3.3.1.7.2

Rewrite the expression.

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

Step 1.5.3.3.2

Combine the numerators over the common denominator.

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

Step 1.5.3.3.3

To write $\frac{- y^{2} - 1}{x y - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.5.3.3.4

Write each expression with a common denominator of $(x y - 1) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.3.4.1

Multiply $\frac{- y^{2} - 1}{x y - 1}$ by $\frac{x}{x}$ .

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

Step 1.5.3.3.4.2

Reorder the factors of $(x y - 1) x$ .

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

Step 1.5.3.3.5

Combine the numerators over the common denominator.

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

Step 1.5.3.3.6

Simplify the numerator.

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Step 1.5.3.3.6.1

Apply the distributive property.

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

Step 1.5.3.3.6.2

Rewrite $- 1 x$ as $- x$ .

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

Step 1.5.3.3.7

Simplify with factoring out.

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Step 1.5.3.3.7.1

Factor $- 1$ out of $- y^{2} x$ .

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

Step 1.5.3.3.7.2

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

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

Step 1.5.3.3.7.3

Factor $- 1$ out of $- (y^{2} x) - (x)$ .

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

Step 1.5.3.3.7.4

Factor $- 1$ out of $y$ .

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

Step 1.5.3.3.7.5

Factor $- 1$ out of $- (y^{2} x + x) - 1 (- y)$ .

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

Step 1.5.3.3.7.6

Simplify the expression.

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Step 1.5.3.3.7.6.1

Rewrite $- (y^{2} x + x - y)$ as $- 1 (y^{2} x + x - y)$ .

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

Step 1.5.3.3.7.6.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = - 1$ and $y = 0$ .

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Step 1.7.1

Replace the variable $x$ with $- 1$ in the expression.

$- \frac{y^{2} (- 1) - 1 - y}{(- 1) ((- 1) \cdot y - 1)}$

Step 1.7.2

Replace the variable $y$ with $0$ in the expression.

$- \frac{{(0)}^{2} (- 1) - 1 - (0)}{(- 1) ((- 1) \cdot (0) - 1)}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Raising $0$ to any positive power yields $0$ .

$- \frac{0 \cdot - 1 - 1 - (0)}{- 1 ((- 1) \cdot (0) - 1)}$

Step 1.7.3.2

Multiply $0$ by $- 1$ .

$- \frac{0 - 1 - (0)}{- 1 ((- 1) \cdot (0) - 1)}$

Step 1.7.3.3

Multiply $- 1$ by $0$ .

$- \frac{0 - 1 + 0}{- 1 ((- 1) \cdot (0) - 1)}$

Step 1.7.3.4

Add $0 - 1$ and $0$ .

$- \frac{0 - 1}{- 1 ((- 1) \cdot (0) - 1)}$

Step 1.7.3.5

Subtract $1$ from $0$ .

$- \frac{- 1}{- 1 ((- 1) \cdot (0) - 1)}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $- 1$ by $0$ .

$- \frac{- 1}{- 1 (0 - 1)}$

Step 1.7.4.2

Subtract $1$ from $0$ .

$- \frac{- 1}{- 1 \cdot - 1}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Multiply $- 1$ by $- 1$ .

$- \frac{- 1}{1}$

Step 1.7.5.2

Divide $- 1$ by $1$ .

$- - 1$

Step 1.7.5.3

Multiply $- 1$ by $- 1$ .

$1$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $1$ and a given point $(- 1, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 1 \cdot (x - (- 1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y + 0 = 1 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 1 \cdot (x + 1)$

Step 2.3.2

Multiply $x + 1$ by $1$ .

$y = x + 1$

Step 3