Calculus Examples

$7 x^{2} + 6 x y + 9 y^{2} + 11 y - 7 = 0$ , $(1, 0)$

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.

$\frac{d}{d x} (7 x^{2} + 6 x y + 9 y^{2} + 11 y - 7) = \frac{d}{d x} (0)$

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 $7 x^{2} + 6 x y + 9 y^{2} + 11 y - 7$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.2

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 1.2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.2.2

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

$7 (2 x) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.2.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.3

Evaluate $\frac{d}{d x} [6 x y]$ .

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Step 1.2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 \frac{d}{d x} [x y]$ .

$14 x + 6 \frac{d}{d x} [x y] + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

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 $g (x) = y$ .

$14 x + 6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.3.3

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

$14 x + 6 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

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 $n = 1$ .

$14 x + 6 (x y' + y \cdot 1) + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.3.5

Multiply $y$ by $1$ .

$14 x + 6 (x y' + y) + \frac{d}{d x} [9 y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4

Evaluate $\frac{d}{d x} [9 y^{2}]$ .

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Step 1.2.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 y^{2}$ with respect to $x$ is $9 \frac{d}{d x} [y^{2}]$ .

$14 x + 6 (x y' + y) + 9 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4.2

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 1.2.4.2.1

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

$14 x + 6 (x y' + y) + 9 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4.2.2

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

$14 x + 6 (x y' + y) + 9 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4.2.3

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

$14 x + 6 (x y' + y) + 9 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4.3

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

$14 x + 6 (x y' + y) + 9 (2 y y') + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.4.4

Multiply $2$ by $9$ .

$14 x + 6 (x y' + y) + 18 y y' + \frac{d}{d x} [11 y] + \frac{d}{d x} [- 7]$

Step 1.2.5

Evaluate $\frac{d}{d x} [11 y]$ .

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Step 1.2.5.1

Since $11$ is constant with respect to $x$ , the derivative of $11 y$ with respect to $x$ is $11 \frac{d}{d x} [y]$ .

$14 x + 6 (x y' + y) + 18 y y' + 11 \frac{d}{d x} [y] + \frac{d}{d x} [- 7]$

Step 1.2.5.2

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

$14 x + 6 (x y' + y) + 18 y y' + 11 y' + \frac{d}{d x} [- 7]$

Step 1.2.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$14 x + 6 (x y' + y) + 18 y y' + 11 y' + 0$

Step 1.2.7

Simplify.

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Step 1.2.7.1

Apply the distributive property.

$14 x + 6 (x y') + 6 y + 18 y y' + 11 y' + 0$

Step 1.2.7.2

Add $14 x + 6 x y' + 6 y + 18 y y' + 11 y'$ and $0$ .

$14 x + 6 x y' + 6 y + 18 y y' + 11 y'$

Step 1.2.7.3

Reorder terms.

$6 x y' + 18 y y' + 14 x + 6 y + 11 y'$

Step 1.3

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

$0$

Step 1.4

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

$6 x y' + 18 y y' + 14 x + 6 y + 11 y' = 0$

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 $14 x$ from both sides of the equation.

$6 x y' + 18 y y' + 6 y + 11 y' = - 14 x$

Step 1.5.1.2

Subtract $6 y$ from both sides of the equation.

$6 x y' + 18 y y' + 11 y' = - 14 x - 6 y$

Step 1.5.2

Factor $y'$ out of $6 x y' + 18 y y' + 11 y'$ .

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Step 1.5.2.1

Factor $y'$ out of $6 x y'$ .

$y' (6 x) + 18 y y' + 11 y' = - 14 x - 6 y$

Step 1.5.2.2

Factor $y'$ out of $18 y y'$ .

$y' (6 x) + y' (18 y) + 11 y' = - 14 x - 6 y$

Step 1.5.2.3

Factor $y'$ out of $11 y'$ .

$y' (6 x) + y' (18 y) + y' \cdot 11 = - 14 x - 6 y$

Step 1.5.2.4

Factor $y'$ out of $y' (6 x) + y' (18 y)$ .

$y' (6 x + 18 y) + y' \cdot 11 = - 14 x - 6 y$

Step 1.5.2.5

Factor $y'$ out of $y' (6 x + 18 y) + y' \cdot 11$ .

$y' (6 x + 18 y + 11) = - 14 x - 6 y$

Step 1.5.3

Divide each term in $y' (6 x + 18 y + 11) = - 14 x - 6 y$ by $6 x + 18 y + 11$ and simplify.

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Step 1.5.3.1

Divide each term in $y' (6 x + 18 y + 11) = - 14 x - 6 y$ by $6 x + 18 y + 11$ .

$\frac{y' (6 x + 18 y + 11)}{6 x + 18 y + 11} = \frac{- 14 x}{6 x + 18 y + 11} + \frac{- 6 y}{6 x + 18 y + 11}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $6 x + 18 y + 11$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{y' (6 x + 18 y + 11)}{6 x + 18 y + 11} = \frac{- 14 x}{6 x + 18 y + 11} + \frac{- 6 y}{6 x + 18 y + 11}$

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 14 x}{6 x + 18 y + 11} + \frac{- 6 y}{6 x + 18 y + 11}$

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.

$y' = \frac{- 14 x - 6 y}{6 x + 18 y + 11}$

Step 1.5.3.3.2

Factor $2$ out of $- 14 x - 6 y$ .

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Step 1.5.3.3.2.1

Factor $2$ out of $- 14 x$ .

$y' = \frac{2 (- 7 x) - 6 y}{6 x + 18 y + 11}$

Step 1.5.3.3.2.2

Factor $2$ out of $- 6 y$ .

$y' = \frac{2 (- 7 x) + 2 (- 3 y)}{6 x + 18 y + 11}$

Step 1.5.3.3.2.3

Factor $2$ out of $2 (- 7 x) + 2 (- 3 y)$ .

$y' = \frac{2 (- 7 x - 3 y)}{6 x + 18 y + 11}$

Step 1.5.3.3.3

Factor $- 1$ out of $- 7 x$ .

$y' = \frac{2 (- (7 x) - 3 y)}{6 x + 18 y + 11}$

Step 1.5.3.3.4

Factor $- 1$ out of $- 3 y$ .

$y' = \frac{2 (- (7 x) - (3 y))}{6 x + 18 y + 11}$

Step 1.5.3.3.5

Factor $- 1$ out of $- (7 x) - (3 y)$ .

$y' = \frac{2 (- (7 x + 3 y))}{6 x + 18 y + 11}$

Step 1.5.3.3.6

Simplify the expression.

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Step 1.5.3.3.6.1

Rewrite $- (7 x + 3 y)$ as $- 1 (7 x + 3 y)$ .

$y' = \frac{2 (- 1 (7 x + 3 y))}{6 x + 18 y + 11}$

Step 1.5.3.3.6.2

Move the negative in front of the fraction.

$y' = - \frac{2 (7 x + 3 y)}{6 x + 18 y + 11}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{2 (7 x + 3 y)}{6 x + 18 y + 11}$

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.

$- \frac{2 (7 (1) + 3 y)}{6 (1) + 18 y + 11}$

Step 1.7.2

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

$- \frac{2 (7 (1) + 3 (0))}{6 (1) + 18 (0) + 11}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Multiply $7$ by $1$ .

$- \frac{2 (7 + 3 (0))}{6 (1) + 18 (0) + 11}$

Step 1.7.3.2

Multiply $3$ by $0$ .

$- \frac{2 (7 + 0)}{6 (1) + 18 (0) + 11}$

Step 1.7.3.3

Add $7$ and $0$ .

$- \frac{2 \cdot 7}{6 (1) + 18 (0) + 11}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $6$ by $1$ .

$- \frac{2 \cdot 7}{6 + 18 (0) + 11}$

Step 1.7.4.2

Multiply $18$ by $0$ .

$- \frac{2 \cdot 7}{6 + 0 + 11}$

Step 1.7.4.3

Add $6$ and $0$ .

$- \frac{2 \cdot 7}{6 + 11}$

Step 1.7.4.4

Add $6$ and $11$ .

$- \frac{2 \cdot 7}{17}$

Step 1.7.5

Multiply $2$ by $7$ .

$- \frac{14}{17}$

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 $- \frac{14}{17}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - \frac{14}{17} \cdot (x - (1))$

Step 2.2

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

$y + 0 = - \frac{14}{17} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = - \frac{14}{17} \cdot (x - 1)$

Step 2.3.2

Simplify $- \frac{14}{17} \cdot (x - 1)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = - \frac{14}{17} x - \frac{14}{17} \cdot - 1$

Step 2.3.2.2

Combine $x$ and $\frac{14}{17}$ .

$y = - \frac{x \cdot 14}{17} - \frac{14}{17} \cdot - 1$

Step 2.3.2.3

Multiply $- \frac{14}{17} \cdot - 1$ .

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Step 2.3.2.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{x \cdot 14}{17} + 1 (\frac{14}{17})$

Step 2.3.2.3.2

Multiply $\frac{14}{17}$ by $1$ .

$y = - \frac{x \cdot 14}{17} + \frac{14}{17}$

Step 2.3.2.4

Move $14$ to the left of $x$ .

$y = - \frac{14 x}{17} + \frac{14}{17}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{14}{17} x) + \frac{14}{17}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{14}{17} x + \frac{14}{17}$

Step 3