Calculus Examples

$2 x y - 2 x + y = - 14$ ; $(2, - 2)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = - 2$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 x y - 2 x + y) = \frac{d}{d x} (- 14)$

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 $2 x y - 2 x + y$ with respect to $x$ is $\frac{d}{d x} [2 x y] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [y]$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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 $g (x) = y$ .

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

Step 1.2.2.3

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

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

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 $n = 1$ .

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

Step 1.2.2.5

Multiply $y$ by $1$ .

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

Multiply $- 2$ by $1$ .

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

Step 1.2.4

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

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

Step 1.2.5

Apply the distributive property.

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

Step 1.3

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

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

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 $2 y$ from both sides of the equation.

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

Step 1.5.1.2

Add $2$ to both sides of the equation.

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Raise $y'$ to the power of $1$ .

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

Step 1.5.2.3

Factor $y'$ out of ${y'}^{1}$ .

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

Step 1.5.2.4

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

$y' (2 x + 1) = - 2 y + 2$

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $2 x + 1$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

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

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.

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

Step 1.5.3.3.2

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

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Step 1.5.3.3.2.1

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

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

Step 1.5.3.3.2.2

Factor $2$ out of $2$ .

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

Step 1.5.3.3.2.3

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

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

Step 1.5.3.3.3

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

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

Step 1.5.3.3.4

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

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

Step 1.5.3.3.5

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

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

Step 1.5.3.3.6

Simplify the expression.

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Step 1.5.3.3.6.1

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

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

Step 1.5.3.3.6.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 2$ and $y = - 2$ .

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Step 1.7.1

Replace the variable $x$ with $2$ in the expression.

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

Step 1.7.2

Replace the variable $y$ with $- 2$ in the expression.

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

Step 1.7.3

Subtract $1$ from $- 2$ .

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

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $2$ by $2$ .

$- \frac{2 \cdot - 3}{4 + 1}$

Step 1.7.4.2

Add $4$ and $1$ .

$- \frac{2 \cdot - 3}{5}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Multiply $2$ by $- 3$ .

$- \frac{- 6}{5}$

Step 1.7.5.2

Move the negative in front of the fraction.

$- - \frac{6}{5}$

Step 1.7.6

Multiply $- - \frac{6}{5}$ .

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Step 1.7.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{6}{5})$

Step 1.7.6.2

Multiply $\frac{6}{5}$ by $1$ .

$\frac{6}{5}$

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 $\frac{6}{5}$ and a given point $(2, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = \frac{6}{5} \cdot (x - (2))$

Step 2.2

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

$y + 2 = \frac{6}{5} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{6}{5} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y + 2 = 0 + 0 + \frac{6}{5} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 2 = \frac{6}{5} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y + 2 = \frac{6}{5} x + \frac{6}{5} \cdot - 2$

Step 2.3.1.4

Combine $\frac{6}{5}$ and $x$ .

$y + 2 = \frac{6 x}{5} + \frac{6}{5} \cdot - 2$

Step 2.3.1.5

Multiply $\frac{6}{5} \cdot - 2$ .

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Step 2.3.1.5.1

Combine $\frac{6}{5}$ and $- 2$ .

$y + 2 = \frac{6 x}{5} + \frac{6 \cdot - 2}{5}$

Step 2.3.1.5.2

Multiply $6$ by $- 2$ .

$y + 2 = \frac{6 x}{5} + \frac{- 12}{5}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y + 2 = \frac{6 x}{5} - \frac{12}{5}$

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

Subtract $2$ from both sides of the equation.

$y = \frac{6 x}{5} - \frac{12}{5} - 2$

Step 2.3.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{6 x}{5} - \frac{12}{5} - 2 \cdot \frac{5}{5}$

Step 2.3.2.3

Combine $- 2$ and $\frac{5}{5}$ .

$y = \frac{6 x}{5} - \frac{12}{5} + \frac{- 2 \cdot 5}{5}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{6 x}{5} + \frac{- 12 - 2 \cdot 5}{5}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 2$ by $5$ .

$y = \frac{6 x}{5} + \frac{- 12 - 10}{5}$

Step 2.3.2.5.2

Subtract $10$ from $- 12$ .

$y = \frac{6 x}{5} + \frac{- 22}{5}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{6 x}{5} - \frac{22}{5}$

Step 2.3.3

Reorder terms.

$y = \frac{6}{5} x - \frac{22}{5}$

Step 3