Calculus Examples

$x^{2} y^{2} = 100$ $(2, - 5)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = - 5$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y^{2}) = \frac{d}{d x} (100)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.3

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Rewrite the expression.

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

Step 1.5.2.2.2

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

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Step 1.5.2.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2.2

Rewrite the expression.

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

Step 1.5.2.2.3

Cancel the common factor of $y$ .

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Step 1.5.2.2.3.1

Cancel the common factor.

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

Step 1.5.2.2.3.2

Divide $y'$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

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

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Step 1.5.2.3.1.1

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

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

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

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

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

Step 1.5.2.3.1.2.2

Cancel the common factor.

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

Step 1.5.2.3.1.2.3

Rewrite the expression.

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

Step 1.5.2.3.2

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

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Step 1.5.2.3.2.1

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

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

Step 1.5.2.3.2.2

Cancel the common factors.

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Step 1.5.2.3.2.2.1

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

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

Step 1.5.2.3.2.2.2

Cancel the common factor.

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

Step 1.5.2.3.2.2.3

Rewrite the expression.

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

Step 1.5.2.3.3

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

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Step 1.5.2.3.3.1

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

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

Step 1.5.2.3.3.2

Cancel the common factors.

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Step 1.5.2.3.3.2.1

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

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

Step 1.5.2.3.3.2.2

Cancel the common factor.

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

Step 1.5.2.3.3.2.3

Rewrite the expression.

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

Step 1.5.2.3.4

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 2$ and $y = - 5$ .

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Step 1.7.1

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

$- \frac{y}{2}$

Step 1.7.2

Replace the variable $y$ with $- 5$ in the expression.

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

Step 1.7.3

Move the negative in front of the fraction.

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

Step 1.7.4

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

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Step 1.7.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{2})$

Step 1.7.4.2

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

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 $- 2$ .

$y + 5 = \frac{5 x}{2} + \frac{5}{2} \cdot (2 (- 1))$

Step 2.3.1.5.2

Cancel the common factor.

$y + 5 = \frac{5 x}{2} + \frac{5}{2} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Rewrite the expression.

$y + 5 = \frac{5 x}{2} + 5 \cdot - 1$

Step 2.3.1.6

Multiply $5$ by $- 1$ .

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

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 $5$ from both sides of the equation.

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

Step 2.3.2.2

Subtract $5$ from $- 5$ .

$y = \frac{5 x}{2} - 10$

Step 2.3.3

Reorder terms.

$y = \frac{5}{2} x - 10$

Step 3