Calculus Examples

$x^{2} y^{3} = y^{2} - 2$ ;, $(1, - 1)$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

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

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 $g (x) = y^{3}$ .

$x^{2} \frac{d}{d x} [y^{3}] + y^{3} \frac{d}{d x} [x^{2}]$

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 $f (x) = x^{3}$ and $g (x) = y$ .

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Step 1.2.2.1

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

$x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x^{2}]$

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 $n = 3$ .

$x^{2} (3 u^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.2.3

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

$x^{2} (3 y^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.3

Move $3$ to the left of $x^{2}$ .

$3 \cdot x^{2} y^{2} \frac{d}{d x} [y] + y^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.4

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

$3 x^{2} y^{2} y' + y^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.5

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

$3 x^{2} y^{2} y' + y^{3} (2 x)$

Step 1.2.6

Move $2$ to the left of $y^{3}$ .

$3 x^{2} y^{2} y' + 2 y^{3} x$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $y^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 2]$ .

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

Step 1.3.2

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

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Step 1.3.2.1

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 1.3.2.1.1

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

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

Step 1.3.2.1.2

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

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

Step 1.3.2.1.3

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

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

Step 1.3.2.2

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

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

Step 1.3.3

Differentiate using the Constant Rule.

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Step 1.3.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 y y' + 0$

Step 1.3.3.2

Add $2 y y'$ and $0$ .

$2 y y'$

Step 1.4

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

$3 x^{2} y^{2} y' + 2 y^{3} x = 2 y y'$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $2 y y'$ from both sides of the equation.

$3 x^{2} y^{2} y' + 2 y^{3} x - 2 y y' = 0$

Step 1.5.2

Subtract $2 y^{3} x$ from both sides of the equation.

$3 x^{2} y^{2} y' - 2 y y' = - 2 y^{3} x$

Step 1.5.3

Factor $y y'$ out of $3 x^{2} y^{2} y' - 2 y y'$ .

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Step 1.5.3.1

Factor $y y'$ out of $3 x^{2} y^{2} y'$ .

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

Step 1.5.3.2

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

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

Step 1.5.3.3

Factor $y y'$ out of $y y' (3 x^{2} y) + y y' \cdot - 2$ .

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

Step 1.5.4

Divide each term in $y y' (3 x^{2} y - 2) = - 2 y^{3} x$ by $y (3 x^{2} y - 2)$ and simplify.

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Step 1.5.4.1

Divide each term in $y y' (3 x^{2} y - 2) = - 2 y^{3} x$ by $y (3 x^{2} y - 2)$ .

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

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $y$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

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

Step 1.5.4.2.1.2

Rewrite the expression.

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

Step 1.5.4.2.2

Cancel the common factor of $3 x^{2} y - 2$ .

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Step 1.5.4.2.2.1

Cancel the common factor.

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

Step 1.5.4.2.2.2

Divide $y'$ by $1$ .

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

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Cancel the common factor of $y^{3}$ and $y$ .

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Step 1.5.4.3.1.1

Factor $y$ out of $- 2 y^{3} x$ .

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

Step 1.5.4.3.1.2

Cancel the common factors.

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Step 1.5.4.3.1.2.1

Cancel the common factor.

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

Step 1.5.4.3.1.2.2

Rewrite the expression.

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

Step 1.5.4.3.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 1$ and $y = - 1$ .

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Step 1.7.1

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

$- \frac{2 y^{2} (1)}{3 {(1)}^{2} y - 2}$

Step 1.7.2

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

$- \frac{2 {(- 1)}^{2} (1)}{3 {(1)}^{2} (- 1) - 2}$

Step 1.7.3

Multiply $2$ by $1$ .

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

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

One to any power is one.

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

Step 1.7.4.2

Multiply $3 \cdot 1 \cdot - 1$ .

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Step 1.7.4.2.1

Multiply $3$ by $1$ .

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

Step 1.7.4.2.2

Multiply $3$ by $- 1$ .

$- \frac{2 {(- 1)}^{2}}{- 3 - 2}$

Step 1.7.4.3

Subtract $2$ from $- 3$ .

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

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Raise $- 1$ to the power of $2$ .

$- \frac{2 \cdot 1}{- 5}$

Step 1.7.5.2

Multiply $2$ by $1$ .

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

Step 1.7.5.3

Move the negative in front of the fraction.

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

Step 1.7.6

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

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Step 1.7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.6.2

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

$\frac{2}{5}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{2}{5} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 1 = 0 + 0 + \frac{2}{5} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Multiply $\frac{2}{5} \cdot - 1$ .

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Step 2.3.1.5.1

Combine $\frac{2}{5}$ and $- 1$ .

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

Step 2.3.1.5.2

Multiply $2$ by $- 1$ .

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

Step 2.3.1.6

Move the negative in front of the fraction.

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

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 $1$ from both sides of the equation.

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

Step 2.3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.3.2.3

Combine $- 1$ and $\frac{5}{5}$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 1$ by $5$ .

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

Step 2.3.2.5.2

Subtract $5$ from $- 2$ .

$y = \frac{2 x}{5} + \frac{- 7}{5}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{2 x}{5} - \frac{7}{5}$

Step 2.3.3

Reorder terms.

$y = \frac{2}{5} x - \frac{7}{5}$

Step 3