Calculus Examples

$x y^{3} - x^{5} y^{2} = - 4$ ; $(- 1, 2)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x y^{3} - x^{5} y^{2}) = \frac{d}{d x} (- 4)$

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 $x y^{3} - x^{5} y^{2}$ with respect to $x$ is $\frac{d}{d x} [x y^{3}] + \frac{d}{d x} [- x^{5} y^{2}]$ .

$\frac{d}{d x} [x y^{3}] + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [x y^{3}]$ .

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Step 1.2.2.1

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

$x \frac{d}{d x} [y^{3}] + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.2

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 1.2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$x (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$x (3 {u_{1}}^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$x (3 y^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.3

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

$x (3 y^{2} y') + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{5} y^{2}]$

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 $n = 1$ .

$x (3 y^{2} y') + y^{3} \cdot 1 + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.5

Move $3$ to the left of $x$ .

$3 \cdot x y^{2} y' + y^{3} \cdot 1 + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.2.6

Multiply $y^{3}$ by $1$ .

$3 x y^{2} y' + y^{3} + \frac{d}{d x} [- x^{5} y^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- x^{5} y^{2}]$ .

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Step 1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5} y^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{5} y^{2}]$ .

$3 x y^{2} y' + y^{3} - \frac{d}{d x} [x^{5} y^{2}]$

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 $f (x) = x^{5}$ and $g (x) = y^{2}$ .

$3 x y^{2} y' + y^{3} - (x^{5} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{5}])$

Step 1.2.3.3

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 1.2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$3 x y^{2} y' + y^{3} - (x^{5} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{5}])$

Step 1.2.3.3.2

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

$3 x y^{2} y' + y^{3} - (x^{5} (2 u_{2} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{5}])$

Step 1.2.3.3.3

Replace all occurrences of $u_{2}$ with $y$ .

$3 x y^{2} y' + y^{3} - (x^{5} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{5}])$

Step 1.2.3.4

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

$3 x y^{2} y' + y^{3} - (x^{5} (2 y y') + y^{2} \frac{d}{d x} [x^{5}])$

Step 1.2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$3 x y^{2} y' + y^{3} - (x^{5} (2 y y') + y^{2} (5 x^{4}))$

Step 1.2.3.6

Move $2$ to the left of $x^{5}$ .

$3 x y^{2} y' + y^{3} - (2 \cdot x^{5} y y' + y^{2} (5 x^{4}))$

Step 1.2.3.7

Move $5$ to the left of $y^{2}$ .

$3 x y^{2} y' + y^{3} - (2 x^{5} y y' + 5 y^{2} x^{4})$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Apply the distributive property.

$3 x y^{2} y' + y^{3} - (2 x^{5} y y') - (5 y^{2} x^{4})$

Step 1.2.4.2

Combine terms.

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Step 1.2.4.2.1

Multiply $2$ by $- 1$ .

$3 x y^{2} y' + y^{3} - 2 (x^{5} y y') - (5 y^{2} x^{4})$

Step 1.2.4.2.2

Multiply $5$ by $- 1$ .

$3 x y^{2} y' + y^{3} - 2 x^{5} y y' - 5 y^{2} x^{4}$

Step 1.2.4.3

Reorder terms.

$y^{3} + 3 y^{2} x y' - 2 x^{5} y y' - 5 x^{4} y^{2}$

Step 1.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$y^{3} + 3 y^{2} x y' - 2 x^{5} y y' - 5 x^{4} y^{2} = 0$

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

$3 y^{2} x y' - 2 x^{5} y y' - 5 x^{4} y^{2} = - y^{3}$

Step 1.5.1.2

Add $5 x^{4} y^{2}$ to both sides of the equation.

$3 y^{2} x y' - 2 x^{5} y y' = - y^{3} + 5 x^{4} y^{2}$

Step 1.5.2

Factor $y x y'$ out of $3 y^{2} x y' - 2 x^{5} y y'$ .

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Step 1.5.2.1

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

$y x y' (3 y) - 2 x^{5} y y' = - y^{3} + 5 x^{4} y^{2}$

Step 1.5.2.2

Factor $y x y'$ out of $- 2 x^{5} y y'$ .

$y x y' (3 y) + y x y' (- 2 x^{4}) = - y^{3} + 5 x^{4} y^{2}$

Step 1.5.2.3

Factor $y x y'$ out of $y x y' (3 y) + y x y' (- 2 x^{4})$ .

$y x y' (3 y - 2 x^{4}) = - y^{3} + 5 x^{4} y^{2}$

Step 1.5.3

Divide each term in $y x y' (3 y - 2 x^{4}) = - y^{3} + 5 x^{4} y^{2}$ by $y x (3 y - 2 x^{4})$ and simplify.

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Step 1.5.3.1

Divide each term in $y x y' (3 y - 2 x^{4}) = - y^{3} + 5 x^{4} y^{2}$ by $y x (3 y - 2 x^{4})$ .

$\frac{y x y' (3 y - 2 x^{4})}{y x (3 y - 2 x^{4})} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $y$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{y x y' (3 y - 2 x^{4})}{y x (3 y - 2 x^{4})} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2.1.2

Rewrite the expression.

$\frac{x y' (3 y - 2 x^{4})}{x (3 y - 2 x^{4})} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2.2

Cancel the common factor of $x$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

$\frac{x y' (3 y - 2 x^{4})}{x (3 y - 2 x^{4})} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2.2.2

Rewrite the expression.

$\frac{y' (3 y - 2 x^{4})}{3 y - 2 x^{4}} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2.3

Cancel the common factor of $3 y - 2 x^{4}$ .

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Step 1.5.3.2.3.1

Cancel the common factor.

$\frac{y' (3 y - 2 x^{4})}{3 y - 2 x^{4}} = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- y^{3}}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Simplify each term.

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Step 1.5.3.3.1.1

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

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Step 1.5.3.3.1.1.1

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

$y' = \frac{y (- y^{2})}{y x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.1.2

Cancel the common factors.

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Step 1.5.3.3.1.1.2.1

Factor $y$ out of $y x (3 y - 2 x^{4})$ .

$y' = \frac{y (- y^{2})}{y (x (3 y - 2 x^{4}))} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{y (- y^{2})}{y (x (3 y - 2 x^{4}))} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{4} y^{2}}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.3

Cancel the common factor of $x^{4}$ and $x$ .

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Step 1.5.3.3.1.3.1

Factor $x$ out of $5 x^{4} y^{2}$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{x (5 x^{3} y^{2})}{y x (3 y - 2 x^{4})}$

Step 1.5.3.3.1.3.2

Cancel the common factors.

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Step 1.5.3.3.1.3.2.1

Factor $x$ out of $y x (3 y - 2 x^{4})$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{x (5 x^{3} y^{2})}{x (y (3 y - 2 x^{4}))}$

Step 1.5.3.3.1.3.2.2

Cancel the common factor.

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{x (5 x^{3} y^{2})}{x (y (3 y - 2 x^{4}))}$

Step 1.5.3.3.1.3.2.3

Rewrite the expression.

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{3} y^{2}}{y (3 y - 2 x^{4})}$

Step 1.5.3.3.1.4

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

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Step 1.5.3.3.1.4.1

Factor $y$ out of $5 x^{3} y^{2}$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{y (5 x^{3} y)}{y (3 y - 2 x^{4})}$

Step 1.5.3.3.1.4.2

Cancel the common factors.

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Step 1.5.3.3.1.4.2.1

Cancel the common factor.

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{y (5 x^{3} y)}{y (3 y - 2 x^{4})}$

Step 1.5.3.3.1.4.2.2

Rewrite the expression.

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{3} y}{3 y - 2 x^{4}}$

Step 1.5.3.3.2

To write $\frac{5 x^{3} y}{3 y - 2 x^{4}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{3} y}{3 y - 2 x^{4}} \cdot \frac{x}{x}$

Step 1.5.3.3.3

Write each expression with a common denominator of $x (3 y - 2 x^{4})$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.3.3.1

Multiply $\frac{5 x^{3} y}{3 y - 2 x^{4}}$ by $\frac{x}{x}$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{3} y x}{(3 y - 2 x^{4}) x}$

Step 1.5.3.3.3.2

Reorder the factors of $(3 y - 2 x^{4}) x$ .

$y' = - \frac{y^{2}}{x (3 y - 2 x^{4})} + \frac{5 x^{3} y x}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- y^{2} + 5 x^{3} y x}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5

Simplify the numerator.

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Step 1.5.3.3.5.1

Factor $y$ out of $- y^{2} + 5 x^{3} y x$ .

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Step 1.5.3.3.5.1.1

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

$y' = \frac{y (- y) + 5 x^{3} y x}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.1.2

Factor $y$ out of $5 x^{3} y x$ .

$y' = \frac{y (- y) + y (5 x^{3} x)}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.1.3

Factor $y$ out of $y (- y) + y (5 x^{3} x)$ .

$y' = \frac{y (- y + 5 x^{3} x)}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.5.3.3.5.2.1

Move $x$ .

$y' = \frac{y (- y + 5 (x \cdot x^{3}))}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.2.2

Multiply $x$ by $x^{3}$ .

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Step 1.5.3.3.5.2.2.1

Raise $x$ to the power of $1$ .

$y' = \frac{y (- y + 5 (x^{1} x^{3}))}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = \frac{y (- y + 5 x^{1 + 3})}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.5.2.3

Add $1$ and $3$ .

$y' = \frac{y (- y + 5 x^{4})}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.6

Simplify with factoring out.

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Step 1.5.3.3.6.1

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

$y' = \frac{y (- (y) + 5 x^{4})}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.6.2

Factor $- 1$ out of $5 x^{4}$ .

$y' = \frac{y (- (y) - (- 5 x^{4}))}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.6.3

Factor $- 1$ out of $- (y) - (- 5 x^{4})$ .

$y' = \frac{y (- (y - 5 x^{4}))}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.6.4

Simplify the expression.

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Step 1.5.3.3.6.4.1

Rewrite $- (y - 5 x^{4})$ as $- 1 (y - 5 x^{4})$ .

$y' = \frac{y (- 1 (y - 5 x^{4}))}{x (3 y - 2 x^{4})}$

Step 1.5.3.3.6.4.2

Move the negative in front of the fraction.

$y' = - \frac{y (y - 5 x^{4})}{x (3 y - 2 x^{4})}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{y (y - 5 x^{4})}{x (3 y - 2 x^{4})}$

Step 1.7

Evaluate at $x = - 1$ and $y = 2$ .

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Step 1.7.1

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

$- \frac{y (y - 5 {(- 1)}^{4})}{(- 1) (3 y - 2 {(- 1)}^{4})}$

Step 1.7.2

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

$- \frac{(2) ((2) - 5 {(- 1)}^{4})}{(- 1) (3 (2) - 2 {(- 1)}^{4})}$

Step 1.7.3

Cancel the common factors.

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Step 1.7.3.1

Factor $2$ out of $(- 1) (3 (2) - 2 {(- 1)}^{4})$ .

$- \frac{(2) ((2) - 5 {(- 1)}^{4})}{2 (- 1 (3 - {(- 1)}^{4}))}$

Step 1.7.3.2

Cancel the common factor.

$- \frac{2 ((2) - 5 {(- 1)}^{4})}{2 (- 1 (3 - {(- 1)}^{4}))}$

Step 1.7.3.3

Rewrite the expression.

$- \frac{(2) - 5 {(- 1)}^{4}}{- 1 (3 - {(- 1)}^{4})}$

Step 1.7.4

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 1.7.4.1

Multiply $- 1$ by ${(- 1)}^{4}$ .

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Step 1.7.4.1.1

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

$- \frac{(2) - 5 {(- 1)}^{4}}{- 1 (3 + {(- 1)}^{1} {(- 1)}^{4})}$

Step 1.7.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{(2) - 5 {(- 1)}^{4}}{- 1 (3 + {(- 1)}^{1 + 4})}$

Step 1.7.4.2

Add $1$ and $4$ .

$- \frac{(2) - 5 {(- 1)}^{4}}{- 1 (3 + {(- 1)}^{5})}$

Step 1.7.5

Simplify the numerator.

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Step 1.7.5.1

Raise $- 1$ to the power of $4$ .

$- \frac{2 - 5 \cdot 1}{- 1 (3 + {(- 1)}^{5})}$

Step 1.7.5.2

Multiply $- 5$ by $1$ .

$- \frac{2 - 5}{- 1 (3 + {(- 1)}^{5})}$

Step 1.7.5.3

Subtract $5$ from $2$ .

$- \frac{- 3}{- 1 (3 + {(- 1)}^{5})}$

Step 1.7.6

Simplify the denominator.

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Step 1.7.6.1

Raise $- 1$ to the power of $5$ .

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

Step 1.7.6.2

Subtract $1$ from $3$ .

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

Step 1.7.7

Reduce the expression by cancelling the common factors.

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Step 1.7.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.7.2

Dividing two negative values results in a positive value.

$- \frac{3}{2}$

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 $- \frac{3}{2}$ and a given point $(- 1, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{3}{2} \cdot (x - (- 1))$

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{3}{2} \cdot (x + 1)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 2 = - \frac{3}{2} x - \frac{3}{2} \cdot 1$

Step 2.3.1.2.2

Combine $x$ and $\frac{3}{2}$ .

$y - 2 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot 1$

Step 2.3.1.2.3

Multiply $- 1$ by $1$ .

$y - 2 = - \frac{x \cdot 3}{2} - \frac{3}{2}$

Step 2.3.1.3

Move $3$ to the left of $x$ .

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

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 $2$ to both sides of the equation.

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $2$ .

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

Step 2.3.2.5.2

Add $- 3$ and $4$ .

$y = - \frac{3 x}{2} + \frac{1}{2}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{1}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{3}{2} x + \frac{1}{2}$

Step 3