Calculus Examples

$x^{2} y^{2} = 16$ , $(- 1, - 4)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = - 4$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y^{2}) = \frac{d}{d x} (16)$

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 $16$ is constant with respect to $x$ , the derivative of $16$ 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 $x = - 1$ and $y = - 4$ .

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Step 1.7.1

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

$- \frac{y}{- 1}$

Step 1.7.2

Replace the variable $y$ with $- 4$ in the expression.

$- \frac{- 4}{- 1}$

Step 1.7.3

Divide $- 4$ by $- 1$ .

$- 1 \cdot 4$

Step 1.7.4

Multiply $- 1$ by $4$ .

$- 4$

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

$y - (- 4) = - 4 \cdot (x - (- 1))$

Step 2.2

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

$y + 4 = - 4 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 4 \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 4 = 0 + 0 - 4 \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 4 = - 4 \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 4 = - 4 x - 4 \cdot 1$

Step 2.3.1.4

Multiply $- 4$ by $1$ .

$y + 4 = - 4 x - 4$

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 $4$ from both sides of the equation.

$y = - 4 x - 4 - 4$

Step 2.3.2.2

Subtract $4$ from $- 4$ .

$y = - 4 x - 8$

Step 3