Calculus Examples

$3 {(x^{2} + y^{2})}^{2} = 100 x y^{2}$ ; $(3, 1)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (100 x y^{2})$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $3 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

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

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

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 1.2.3

Differentiate.

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Step 1.2.3.1

Multiply $2$ by $3$ .

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.4

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 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

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

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

Step 1.2.5

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

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

Step 1.2.6

Simplify.

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Step 1.2.6.1

Apply the distributive property.

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

Step 1.2.6.2

Reorder the factors of $(6 x^{2} + 6 y^{2}) (2 x + 2 y y')$ .

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

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since $100$ is constant with respect to $x$ , the derivative of $100 x y^{2}$ with respect to $x$ is $100 \frac{d}{d x} [x y^{2}]$ .

$100 \frac{d}{d x} [x y^{2}]$

Step 1.3.2

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

$100 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

Step 1.3.3

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 1.3.3.1

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

$100 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.2

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

$100 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.3

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

$100 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.4

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

$100 (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Step 1.3.5

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

$100 (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 1.3.6

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

$100 (2 x y y' + y^{2} \cdot 1)$

Step 1.3.7

Multiply $y^{2}$ by $1$ .

$100 (2 x y y' + y^{2})$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Apply the distributive property.

$100 (2 x y y') + 100 y^{2}$

Step 1.3.8.2

Multiply $2$ by $100$ .

$200 x y y' + 100 y^{2}$

Step 1.3.8.3

Reorder terms.

$100 y^{2} + 200 x y y'$

Step 1.4

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

$(2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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Step 1.5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (6 x^{2} + 6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.3

Expand $(2 x + 2 y y') (6 x^{2} + 6 y^{2})$ using the FOIL Method.

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Step 1.5.1.3.1

Apply the distributive property.

$2 x (6 x^{2} + 6 y^{2}) + 2 y y' (6 x^{2} + 6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.3.2

Apply the distributive property.

$2 x (6 x^{2}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2} + 6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.3.3

Apply the distributive property.

$2 x (6 x^{2}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4

Simplify each term.

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Step 1.5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 6 x \cdot x^{2} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.5.1.4.2.1

Move $x^{2}$ .

$2 \cdot 6 (x^{2} x) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.5.1.4.2.2.1

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

$2 \cdot 6 (x^{2} x^{1}) + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.2.2.2

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

$2 \cdot 6 x^{2 + 1} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 6 x^{3} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.3

Multiply $2$ by $6$ .

$12 x^{3} + 2 x (6 y^{2}) + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.4

Rewrite using the commutative property of multiplication.

$12 x^{3} + 2 \cdot 6 x y^{2} + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.5

Multiply $2$ by $6$ .

$12 x^{3} + 12 x y^{2} + 2 y y' (6 x^{2}) + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.6

Multiply $6$ by $2$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y y' (6 y^{2}) = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.7

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 1.5.1.4.7.1

Move $y^{2}$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 (y^{2} y) y' \cdot 6 = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.7.2

Multiply $y^{2}$ by $y$ .

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Step 1.5.1.4.7.2.1

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

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 6 = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.7.2.2

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

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y^{2 + 1} y' \cdot 6 = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.7.3

Add $2$ and $1$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 2 y^{3} y' \cdot 6 = 100 y^{2} + 200 x y y'$

Step 1.5.1.4.8

Multiply $6$ by $2$ .

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' = 100 y^{2} + 200 x y y'$

Step 1.5.2

Subtract $200 x y y'$ from both sides of the equation.

$12 x^{3} + 12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' - 200 x y y' = 100 y^{2}$

Step 1.5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 1.5.3.1

Subtract $12 x^{3}$ from both sides of the equation.

$12 x y^{2} + 12 y y' x^{2} + 12 y^{3} y' - 200 x y y' = 100 y^{2} - 12 x^{3}$

Step 1.5.3.2

Subtract $12 x y^{2}$ from both sides of the equation.

$12 y y' x^{2} + 12 y^{3} y' - 200 x y y' = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.4

Factor $4 y y'$ out of $12 y y' x^{2} + 12 y^{3} y' - 200 x y y'$ .

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Step 1.5.4.1

Factor $4 y y'$ out of $12 y y' x^{2}$ .

$4 y y' (3 x^{2}) + 12 y^{3} y' - 200 x y y' = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.2

Factor $4 y y'$ out of $12 y^{3} y'$ .

$4 y y' (3 x^{2}) + 4 y y' (3 y^{2}) - 200 x y y' = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.3

Factor $4 y y'$ out of $- 200 x y y'$ .

$4 y y' (3 x^{2}) + 4 y y' (3 y^{2}) + 4 y y' (- 50 x) = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.4

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

$4 y y' (3 x^{2} + 3 y^{2}) + 4 y y' (- 50 x) = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.5

Factor $4 y y'$ out of $4 y y' (3 x^{2} + 3 y^{2}) + 4 y y' (- 50 x)$ .

$4 y y' (3 x^{2} + 3 y^{2} - 50 x) = 100 y^{2} - 12 x^{3} - 12 x y^{2}$

Step 1.5.5

Divide each term in $4 y y' (3 x^{2} + 3 y^{2} - 50 x) = 100 y^{2} - 12 x^{3} - 12 x y^{2}$ by $4 y (3 x^{2} + 3 y^{2} - 50 x)$ and simplify.

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Step 1.5.5.1

Divide each term in $4 y y' (3 x^{2} + 3 y^{2} - 50 x) = 100 y^{2} - 12 x^{3} - 12 x y^{2}$ by $4 y (3 x^{2} + 3 y^{2} - 50 x)$ .

$\frac{4 y y' (3 x^{2} + 3 y^{2} - 50 x)}{4 y (3 x^{2} + 3 y^{2} - 50 x)} = \frac{100 y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $4$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

Step 1.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (3 x^{2} + 3 y^{2} - 50 x)}{y (3 x^{2} + 3 y^{2} - 50 x)} = \frac{100 y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.2.2

Cancel the common factor of $y$ .

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Step 1.5.5.2.2.1

Cancel the common factor.

Step 1.5.5.2.2.2

Rewrite the expression.

$\frac{y' (3 x^{2} + 3 y^{2} - 50 x)}{3 x^{2} + 3 y^{2} - 50 x} = \frac{100 y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.2.3

Cancel the common factor of $3 x^{2} + 3 y^{2} - 50 x$ .

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Step 1.5.5.2.3.1

Cancel the common factor.

Step 1.5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{100 y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Simplify each term.

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Step 1.5.5.3.1.1

Cancel the common factor of $100$ and $4$ .

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Step 1.5.5.3.1.1.1

Factor $4$ out of $100 y^{2}$ .

$y' = \frac{4 (25 y^{2})}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.1.2

Cancel the common factors.

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Step 1.5.5.3.1.1.2.1

Factor $4$ out of $4 y (3 x^{2} + 3 y^{2} - 50 x)$ .

$y' = \frac{4 (25 y^{2})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{4 (25 y^{2})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{25 y^{2}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.2

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

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Step 1.5.5.3.1.2.1

Factor $y$ out of $25 y^{2}$ .

$y' = \frac{y (25 y)}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.2.2

Cancel the common factors.

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Step 1.5.5.3.1.2.2.1

Cancel the common factor.

$y' = \frac{y (25 y)}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.2.2.2

Rewrite the expression.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.3

Cancel the common factor of $- 12$ and $4$ .

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Step 1.5.5.3.1.3.1

Factor $4$ out of $- 12 x^{3}$ .

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} + \frac{4 (- 3 x^{3})}{4 y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.3.2

Cancel the common factors.

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Step 1.5.5.3.1.3.2.1

Factor $4$ out of $4 y (3 x^{2} + 3 y^{2} - 50 x)$ .

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} + \frac{4 (- 3 x^{3})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.3.2.2

Cancel the common factor.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} + \frac{4 (- 3 x^{3})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.3.2.3

Rewrite the expression.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} + \frac{- 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.4

Move the negative in front of the fraction.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.5

Cancel the common factor of $- 12$ and $4$ .

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Step 1.5.5.3.1.5.1

Factor $4$ out of $- 12 x y^{2}$ .

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{4 (- 3 x y^{2})}{4 y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.5.2

Cancel the common factors.

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Step 1.5.5.3.1.5.2.1

Factor $4$ out of $4 y (3 x^{2} + 3 y^{2} - 50 x)$ .

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{4 (- 3 x y^{2})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))}$

Step 1.5.5.3.1.5.2.2

Cancel the common factor.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{4 (- 3 x y^{2})}{4 (y (3 x^{2} + 3 y^{2} - 50 x))}$

Step 1.5.5.3.1.5.2.3

Rewrite the expression.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 3 x y^{2}}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.6

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

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Step 1.5.5.3.1.6.1

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

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{y (- 3 x y)}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.6.2

Cancel the common factors.

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Step 1.5.5.3.1.6.2.1

Cancel the common factor.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{y (- 3 x y)}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.1.6.2.2

Rewrite the expression.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} + \frac{- 3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.1.7

Move the negative in front of the fraction.

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.2

To write $\frac{25 y}{3 x^{2} + 3 y^{2} - 50 x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = \frac{25 y}{3 x^{2} + 3 y^{2} - 50 x} \cdot \frac{y}{y} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.3

Write each expression with a common denominator of $(3 x^{2} + 3 y^{2} - 50 x) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.5.3.3.1

Multiply $\frac{25 y}{3 x^{2} + 3 y^{2} - 50 x}$ by $\frac{y}{y}$ .

$y' = \frac{25 y \cdot y}{(3 x^{2} + 3 y^{2} - 50 x) y} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.3.2

Reorder the factors of $(3 x^{2} + 3 y^{2} - 50 x) y$ .

$y' = \frac{25 y \cdot y}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.4

Combine the numerators over the common denominator.

$y' = \frac{25 y \cdot y - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.5

Multiply $y$ by $y$ by adding the exponents.

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Step 1.5.5.3.5.1

Move $y$ .

$y' = \frac{25 (y \cdot y) - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.5.2

Multiply $y$ by $y$ .

$y' = \frac{25 y^{2} - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$

Step 1.5.5.3.6

To write $- \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = \frac{25 y^{2} - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x} \cdot \frac{y}{y}$

Step 1.5.5.3.7

Write each expression with a common denominator of $y (3 x^{2} + 3 y^{2} - 50 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.5.3.7.1

Multiply $\frac{3 x y}{3 x^{2} + 3 y^{2} - 50 x}$ by $\frac{y}{y}$ .

$y' = \frac{25 y^{2} - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y \cdot y}{(3 x^{2} + 3 y^{2} - 50 x) y}$

Step 1.5.5.3.7.2

Reorder the factors of $(3 x^{2} + 3 y^{2} - 50 x) y$ .

$y' = \frac{25 y^{2} - 3 x^{3}}{y (3 x^{2} + 3 y^{2} - 50 x)} - \frac{3 x y \cdot y}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.8

Combine the numerators over the common denominator.

$y' = \frac{25 y^{2} - 3 x^{3} - 3 x y \cdot y}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.9

Multiply $y$ by $y$ by adding the exponents.

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Step 1.5.5.3.9.1

Move $y$ .

$y' = \frac{25 y^{2} - 3 x^{3} - 3 x (y \cdot y)}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.5.5.3.9.2

Multiply $y$ by $y$ .

$y' = \frac{25 y^{2} - 3 x^{3} - 3 x y^{2}}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.6

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

$\frac{d y}{d x} = \frac{25 y^{2} - 3 x^{3} - 3 x y^{2}}{y (3 x^{2} + 3 y^{2} - 50 x)}$

Step 1.7

Evaluate at $x = 3$ and $y = 1$ .

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Step 1.7.1

Replace the variable $x$ with $3$ in the expression.

$\frac{25 y^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot y^{2}}{y (3 {(3)}^{2} + 3 y^{2} - 50 \cdot 3)}$

Step 1.7.2

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

$\frac{25 {(1)}^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot {(1)}^{2}}{(1) (3 {(3)}^{2} + 3 {(1)}^{2} - 50 \cdot 3)}$

Step 1.7.3

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 1.7.3.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 1.7.3.1.1

Raise $3$ to the power of $1$ .

$\frac{25 {(1)}^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot {(1)}^{2}}{1 (3^{1} {(3)}^{2} + 3 {(1)}^{2} - 50 \cdot 3)}$

Step 1.7.3.1.2

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

$\frac{25 {(1)}^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot {(1)}^{2}}{1 (3^{1 + 2} + 3 {(1)}^{2} - 50 \cdot 3)}$

Step 1.7.3.2

Add $1$ and $2$ .

$\frac{25 {(1)}^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot {(1)}^{2}}{1 (3^{3} + 3 {(1)}^{2} - 50 \cdot 3)}$

Step 1.7.4

Multiply $3^{3} + 3 {(1)}^{2} - 50 \cdot 3$ by $1$ .

$\frac{25 {(1)}^{2} - 3 {(3)}^{3} - 3 \cdot 3 \cdot {(1)}^{2}}{3^{3} + 3 {(1)}^{2} - 50 \cdot 3}$

Step 1.7.5

Simplify the numerator.

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Step 1.7.5.1

One to any power is one.

$\frac{25 \cdot 1 - 3 \cdot 3^{3} - 3 \cdot 3 \cdot 1^{2}}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.2

Multiply $25$ by $1$ .

$\frac{25 - 3 \cdot 3^{3} - 3 \cdot 3 \cdot 1^{2}}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.3

Raise $3$ to the power of $3$ .

$\frac{25 - 3 \cdot 27 - 3 \cdot 3 \cdot 1^{2}}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.4

Multiply $- 3$ by $27$ .

$\frac{25 - 81 - 3 \cdot 3 \cdot 1^{2}}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.5

Multiply $- 3$ by $3$ .

$\frac{25 - 81 - 9 \cdot 1^{2}}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.6

One to any power is one.

$\frac{25 - 81 - 9 \cdot 1}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.7

Multiply $- 9$ by $1$ .

$\frac{25 - 81 - 9}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.8

Subtract $81$ from $25$ .

$\frac{- 56 - 9}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.5.9

Subtract $9$ from $- 56$ .

$\frac{- 65}{3^{3} + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.6

Simplify the denominator.

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Step 1.7.6.1

Raise $3$ to the power of $3$ .

$\frac{- 65}{27 + 3 \cdot 1^{2} - 50 \cdot 3}$

Step 1.7.6.2

One to any power is one.

$\frac{- 65}{27 + 3 \cdot 1 - 50 \cdot 3}$

Step 1.7.6.3

Multiply $3$ by $1$ .

$\frac{- 65}{27 + 3 - 50 \cdot 3}$

Step 1.7.6.4

Multiply $- 50$ by $3$ .

$\frac{- 65}{27 + 3 - 150}$

Step 1.7.6.5

Add $27$ and $3$ .

$\frac{- 65}{30 - 150}$

Step 1.7.6.6

Subtract $150$ from $30$ .

$\frac{- 65}{- 120}$

Step 1.7.7

Cancel the common factor of $- 65$ and $- 120$ .

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Step 1.7.7.1

Factor $- 5$ out of $- 65$ .

$\frac{- 5 (13)}{- 120}$

Step 1.7.7.2

Cancel the common factors.

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Step 1.7.7.2.1

Factor $- 5$ out of $- 120$ .

$\frac{- 5 \cdot 13}{- 5 \cdot 24}$

Step 1.7.7.2.2

Cancel the common factor.

$\frac{- 5 \cdot 13}{- 5 \cdot 24}$

Step 1.7.7.2.3

Rewrite the expression.

$\frac{13}{24}$

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 $\frac{13}{24}$ and a given point $(3, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{13}{24} \cdot (x - (3))$

Step 2.2

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

$y - 1 = \frac{13}{24} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{13}{24} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{13}{24} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = \frac{13}{24} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = \frac{13}{24} x + \frac{13}{24} \cdot - 3$

Step 2.3.1.4

Combine $\frac{13}{24}$ and $x$ .

$y - 1 = \frac{13 x}{24} + \frac{13}{24} \cdot - 3$

Step 2.3.1.5

Cancel the common factor of $3$ .

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Step 2.3.1.5.1

Factor $3$ out of $24$ .

$y - 1 = \frac{13 x}{24} + \frac{13}{3 (8)} \cdot - 3$

Step 2.3.1.5.2

Factor $3$ out of $- 3$ .

$y - 1 = \frac{13 x}{24} + \frac{13}{3 \cdot 8} \cdot (3 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 1 = \frac{13 x}{24} + \frac{13}{3 \cdot 8} \cdot (3 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 1 = \frac{13 x}{24} + \frac{13}{8} \cdot - 1$

Step 2.3.1.6

Combine $\frac{13}{8}$ and $- 1$ .

$y - 1 = \frac{13 x}{24} + \frac{13 \cdot - 1}{8}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $13$ by $- 1$ .

$y - 1 = \frac{13 x}{24} + \frac{- 13}{8}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - 1 = \frac{13 x}{24} - \frac{13}{8}$

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 $1$ to both sides of the equation.

$y = \frac{13 x}{24} - \frac{13}{8} + 1$

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

$y = \frac{13 x}{24} - \frac{13}{8} + \frac{8}{8}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = \frac{13 x}{24} + \frac{- 13 + 8}{8}$

Step 2.3.2.4

Add $- 13$ and $8$ .

$y = \frac{13 x}{24} + \frac{- 5}{8}$

Step 2.3.2.5

Move the negative in front of the fraction.

$y = \frac{13 x}{24} - \frac{5}{8}$

Step 2.3.3

Reorder terms.

$y = \frac{13}{24} x - \frac{5}{8}$

Step 3