Calculus Examples

$y^{2} - x^{2} y - 6 x = 0$ , $(2, 6)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 6$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

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

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

Step 1.2.2

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

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Step 1.2.2.1

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 1.2.2.1.1

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

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2} y] + \frac{d}{d x} [- 6 x]$

Step 1.2.2.1.2

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

$2 u \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2} y] + \frac{d}{d x} [- 6 x]$

Step 1.2.2.1.3

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

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

Step 1.2.2.2

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

$2 y y' + \frac{d}{d x} [- x^{2} y] + \frac{d}{d x} [- 6 x]$

Step 1.2.3

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

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Step 1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2} y$ with respect to $x$ is $- \frac{d}{d x} [x^{2} y]$ .

$2 y y' - \frac{d}{d x} [x^{2} y] + \frac{d}{d x} [- 6 x]$

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

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

Step 1.2.3.3

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

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

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 $n = 2$ .

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

Step 1.2.3.5

Move $2$ to the left of $y$ .

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

$2 y y' - (x^{2} y' + 2 y x) - 6 \cdot 1$

Step 1.2.4.3

Multiply $- 6$ by $1$ .

$2 y y' - (x^{2} y' + 2 y x) - 6$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Apply the distributive property.

$2 y y' - (x^{2} y') - (2 y x) - 6$

Step 1.2.5.2

Multiply $2$ by $- 1$ .

$2 y y' - x^{2} y' - 2 y x - 6$

Step 1.2.5.3

Reorder terms.

$2 y y' - x^{2} y' - 2 x y - 6$

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.

$2 y y' - x^{2} y' - 2 x y - 6 = 0$

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

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

$2 y y' - x^{2} y' - 6 = 2 x y$

Step 1.5.1.2

Add $6$ to both sides of the equation.

$2 y y' - x^{2} y' = 2 x y + 6$

Step 1.5.2

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

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Step 1.5.2.1

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

$y' (2 y) - x^{2} y' = 2 x y + 6$

Step 1.5.2.2

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

$y' (2 y) + y' (- x^{2}) = 2 x y + 6$

Step 1.5.2.3

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

$y' (2 y - x^{2}) = 2 x y + 6$

Step 1.5.3

Divide each term in $y' (2 y - x^{2}) = 2 x y + 6$ by $2 y - x^{2}$ and simplify.

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Step 1.5.3.1

Divide each term in $y' (2 y - x^{2}) = 2 x y + 6$ by $2 y - x^{2}$ .

$\frac{y' (2 y - x^{2})}{2 y - x^{2}} = \frac{2 x y}{2 y - x^{2}} + \frac{6}{2 y - x^{2}}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

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

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{y' (2 y - x^{2})}{2 y - x^{2}} = \frac{2 x y}{2 y - x^{2}} + \frac{6}{2 y - x^{2}}$

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x y}{2 y - x^{2}} + \frac{6}{2 y - x^{2}}$

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.

$y' = \frac{2 x y + 6}{2 y - x^{2}}$

Step 1.5.3.3.2

Factor $2$ out of $2 x y + 6$ .

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Step 1.5.3.3.2.1

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

$y' = \frac{2 (x y) + 6}{2 y - x^{2}}$

Step 1.5.3.3.2.2

Factor $2$ out of $6$ .

$y' = \frac{2 (x y) + 2 \cdot 3}{2 y - x^{2}}$

Step 1.5.3.3.2.3

Factor $2$ out of $2 (x y) + 2 \cdot 3$ .

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 2$ and $y = 6$ .

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Step 1.7.1

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

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

Step 1.7.2

Replace the variable $y$ with $6$ in the expression.

$\frac{2 ((2) \cdot (6) + 3)}{2 (6) - {(2)}^{2}}$

Step 1.7.3

Cancel the common factor of $2$ and $2 (6) - {(2)}^{2}$ .

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Step 1.7.3.1

Reorder terms.

$\frac{2 ((2) \cdot (6) + 3)}{- {(2)}^{2} + 2 \cdot 6}$

Step 1.7.3.2

Cancel the common factors.

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Step 1.7.3.2.1

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

$\frac{2 ((2) \cdot (6) + 3)}{2 (- 1 \cdot 2) + 2 \cdot 6}$

Step 1.7.3.2.2

Factor $2$ out of $2 \cdot 6$ .

$\frac{2 ((2) \cdot (6) + 3)}{2 (- 1 \cdot 2) + 2 (6)}$

Step 1.7.3.2.3

Factor $2$ out of $2 (- 1 \cdot 2) + 2 (6)$ .

$\frac{2 ((2) \cdot (6) + 3)}{2 (- 1 \cdot 2 + 6)}$

Step 1.7.3.2.4

Cancel the common factor.

$\frac{2 ((2) \cdot (6) + 3)}{2 (- 1 \cdot 2 + 6)}$

Step 1.7.3.2.5

Rewrite the expression.

$\frac{(2) \cdot (6) + 3}{- 1 \cdot 2 + 6}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Multiply $2$ by $6$ .

$\frac{12 + 3}{- 1 \cdot 2 + 6}$

Step 1.7.4.2

Add $12$ and $3$ .

$\frac{15}{- 1 \cdot 2 + 6}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Multiply $- 1$ by $2$ .

$\frac{15}{- 2 + 6}$

Step 1.7.5.2

Add $- 2$ and $6$ .

$\frac{15}{4}$

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 $\frac{15}{4}$ and a given point $(2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{15}{4} \cdot (x - (2))$

Step 2.2

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

$y - 6 = \frac{15}{4} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{15}{4} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 + \frac{15}{4} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = \frac{15}{4} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = \frac{15}{4} x + \frac{15}{4} \cdot - 2$

Step 2.3.1.4

Combine $\frac{15}{4}$ and $x$ .

$y - 6 = \frac{15 x}{4} + \frac{15}{4} \cdot - 2$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $4$ .

$y - 6 = \frac{15 x}{4} + \frac{15}{2 (2)} \cdot - 2$

Step 2.3.1.5.2

Factor $2$ out of $- 2$ .

$y - 6 = \frac{15 x}{4} + \frac{15}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 6 = \frac{15 x}{4} + \frac{15}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 6 = \frac{15 x}{4} + \frac{15}{2} \cdot - 1$

Step 2.3.1.6

Combine $\frac{15}{2}$ and $- 1$ .

$y - 6 = \frac{15 x}{4} + \frac{15 \cdot - 1}{2}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $15$ by $- 1$ .

$y - 6 = \frac{15 x}{4} + \frac{- 15}{2}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - 6 = \frac{15 x}{4} - \frac{15}{2}$

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 $6$ to both sides of the equation.

$y = \frac{15 x}{4} - \frac{15}{2} + 6$

Step 2.3.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{15 x}{4} - \frac{15}{2} + 6 \cdot \frac{2}{2}$

Step 2.3.2.3

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

$y = \frac{15 x}{4} - \frac{15}{2} + \frac{6 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{15 x}{4} + \frac{- 15 + 6 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $6$ by $2$ .

$y = \frac{15 x}{4} + \frac{- 15 + 12}{2}$

Step 2.3.2.5.2

Add $- 15$ and $12$ .

$y = \frac{15 x}{4} + \frac{- 3}{2}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{15 x}{4} - \frac{3}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{15}{4} x - \frac{3}{2}$

Step 3