Calculus Examples

$x^{2} + x y + y^{2} = 3$ , $(1, 1)$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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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 $f (x) = x$ and $g (x) = y$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.2.4

Multiply $y$ by $1$ .

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

Step 1.2.3

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

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Step 1.2.3.1

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 1.2.3.1.1

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

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

Step 1.2.3.1.2

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

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

Step 1.2.3.1.3

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

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

Step 1.2.3.2

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

$2 x + x y' + y + 2 y y'$

Step 1.2.4

Reorder terms.

$x y' + 2 y y' + 2 x + y$

Step 1.3

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

$0$

Step 1.4

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

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

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 $2 x$ from both sides of the equation.

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

Step 1.5.1.2

Subtract $y$ from both sides of the equation.

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

$y' (x) + y' (2 y) = - 2 x - y$

Step 1.5.2.3

Factor $y'$ out of $y' (x) + y' (2 y)$ .

$y' (x + 2 y) = - 2 x - y$

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $x + 2 y$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Combine the numerators over the common denominator.

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

Step 1.5.3.3.2

Factor $- 1$ out of $- 2 x$ .

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

Step 1.5.3.3.3

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

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

Step 1.5.3.3.4

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

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

Step 1.5.3.3.5

Simplify the expression.

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Step 1.5.3.3.5.1

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

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

Step 1.5.3.3.5.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

$- \frac{2 (1) + y}{(1) + 2 y}$

Step 1.7.2

Replace the variable $y$ with $1$ in the expression.

$- \frac{2 (1) + 1}{(1) + 2 (1)}$

Step 1.7.3

Remove parentheses.

$- \frac{2 (1) + 1}{(1) + 2 (1)}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Multiply $2$ by $1$ .

$- \frac{2 + 1}{1 + 2 (1)}$

Step 1.7.4.2

Add $2$ and $1$ .

$- \frac{3}{1 + 2 (1)}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Multiply $2$ by $1$ .

$- \frac{3}{1 + 2}$

Step 1.7.5.2

Add $1$ and $2$ .

$- \frac{3}{3}$

Step 1.7.6

Reduce the expression by cancelling the common factors.

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Step 1.7.6.1

Cancel the common factor of $3$ .

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Step 1.7.6.1.1

Cancel the common factor.

$- \frac{3}{3}$

Step 1.7.6.1.2

Rewrite the expression.

$- 1 \cdot 1$

Step 1.7.6.2

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 $(1, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

$y - 1 = - 1 x - 1 \cdot - 1$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

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

$y - 1 = - x - 1 \cdot - 1$

Step 2.3.1.4.2

Multiply $- 1$ by $- 1$ .

$y - 1 = - x + 1$

Step 2.3.2

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

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Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = - x + 1 + 1$

Step 2.3.2.2

Add $1$ and $1$ .

$y = - x + 2$

Step 3