Calculus Examples

$3 {(x^{2} + y^{2})}^{2} = 100 (x^{2} - y^{2})$ , $(4, 2)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

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

Differentiate.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.1.4

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

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

Step 1.3.2

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 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

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

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

Step 1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.3.4

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

$100 (2 x - 2 y y')$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Apply the distributive property.

$100 (2 x) + 100 (- 2 y y')$

Step 1.3.5.2

Combine terms.

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Step 1.3.5.2.1

Multiply $2$ by $100$ .

$200 x + 100 (- 2 y y')$

Step 1.3.5.2.2

Multiply $- 2$ by $100$ .

$200 x - 200 y y'$

Step 1.3.5.3

Reorder terms.

$- 200 y y' + 200 x$

Step 1.4

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

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

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.

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

Step 1.5.1.2

Simplify by adding zeros.

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

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.

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

Step 1.5.1.3.2

Apply the distributive property.

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

Step 1.5.1.3.3

Apply the distributive property.

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

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.

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

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 $x^{2}$ .

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

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 $1$ .

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

Step 1.5.1.4.2.2.2

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

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

Step 1.5.1.4.2.3

Add $2$ and $1$ .

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

Step 1.5.1.4.3

Multiply $2$ by $6$ .

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

Step 1.5.1.4.4

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.4.5

Multiply $2$ by $6$ .

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

Step 1.5.1.4.6

Multiply $6$ by $2$ .

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

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 $y^{2}$ .

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

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 $1$ .

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

Step 1.5.1.4.7.2.2

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

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

Step 1.5.1.4.7.3

Add $2$ and $1$ .

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

Step 1.5.1.4.8

Multiply $6$ by $2$ .

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

Step 1.5.2

Add $200 y y'$ to both sides of the equation.

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

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.

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

Step 1.5.3.2

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

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

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

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

Step 1.5.4.3

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

$4 y y' (3 x^{2}) + 4 y y' (3 y^{2}) + 4 y y' \cdot 50 = 200 x - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.4

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

$4 y y' (3 x^{2} + 3 y^{2}) + 4 y y' \cdot 50 = 200 x - 12 x^{3} - 12 x y^{2}$

Step 1.5.4.5

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

$4 y y' (3 x^{2} + 3 y^{2} + 50) = 200 x - 12 x^{3} - 12 x y^{2}$

Step 1.5.5

Divide each term in $4 y y' (3 x^{2} + 3 y^{2} + 50) = 200 x - 12 x^{3} - 12 x y^{2}$ by $4 y (3 x^{2} + 3 y^{2} + 50)$ and simplify.

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Step 1.5.5.1

Divide each term in $4 y y' (3 x^{2} + 3 y^{2} + 50) = 200 x - 12 x^{3} - 12 x y^{2}$ by $4 y (3 x^{2} + 3 y^{2} + 50)$ .

$\frac{4 y y' (3 x^{2} + 3 y^{2} + 50)}{4 y (3 x^{2} + 3 y^{2} + 50)} = \frac{200 x}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} + 50)}$

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.

$\frac{y y' (3 x^{2} + 3 y^{2} + 50)}{y (3 x^{2} + 3 y^{2} + 50)} = \frac{200 x}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} + 50)}$

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.

$\frac{y' (3 x^{2} + 3 y^{2} + 50)}{3 x^{2} + 3 y^{2} + 50} = \frac{200 x}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} + 50)}$

Step 1.5.5.2.3

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

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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 $1$ .

$y' = \frac{200 x}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x^{3}}{4 y (3 x^{2} + 3 y^{2} + 50)} + \frac{- 12 x y^{2}}{4 y (3 x^{2} + 3 y^{2} + 50)}$

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 $200$ and $4$ .

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Step 1.5.5.3.1.1.1

Factor $4$ out of $200 x$ .

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

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 $4 y (3 x^{2} + 3 y^{2} + 50)$ .

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

Step 1.5.5.3.1.1.2.2

Cancel the common factor.

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

Step 1.5.5.3.1.1.2.3

Rewrite the expression.

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

Step 1.5.5.3.1.2

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

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Step 1.5.5.3.1.2.1

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

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

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

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

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

Step 1.5.5.3.1.2.2.2

Cancel the common factor.

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

Step 1.5.5.3.1.2.2.3

Rewrite the expression.

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

Step 1.5.5.3.1.3

Move the negative in front of the fraction.

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

Step 1.5.5.3.1.4

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

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Step 1.5.5.3.1.4.1

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

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

Step 1.5.5.3.1.4.2

Cancel the common factors.

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Step 1.5.5.3.1.4.2.1

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

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

Step 1.5.5.3.1.4.2.2

Cancel the common factor.

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

Step 1.5.5.3.1.4.2.3

Rewrite the expression.

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

Step 1.5.5.3.1.5

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

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Step 1.5.5.3.1.5.1

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

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

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

Cancel the common factor.

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

Step 1.5.5.3.1.5.2.2

Rewrite the expression.

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

Step 1.5.5.3.1.6

Move the negative in front of the fraction.

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

Step 1.5.5.3.2

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

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

Step 1.5.5.3.3

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

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Step 1.5.5.3.3.1

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

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

Step 1.5.5.3.3.2

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

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

Step 1.5.5.3.4

Combine the numerators over the common denominator.

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

Step 1.5.5.3.5

Combine the numerators over the common denominator.

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

Step 1.5.5.3.6

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

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Step 1.5.5.3.6.1

Move $y$ .

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

Step 1.5.5.3.6.2

Multiply $y$ by $y$ .

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

Step 1.5.5.3.7

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

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Step 1.5.5.3.7.1

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

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

Step 1.5.5.3.7.2

Factor $x$ out of $50 x$ .

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

Step 1.5.5.3.7.3

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

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

Step 1.5.5.3.7.4

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

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

Step 1.5.5.3.7.5

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

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

Step 1.5.5.3.8

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 1.5.5.3.9

Rewrite $50$ as $- 1 (- 50)$ .

$y' = \frac{x (- (3 x^{2}) - 1 (- 50) - 3 y^{2})}{y (3 x^{2} + 3 y^{2} + 50)}$

Step 1.5.5.3.10

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 50)$ .

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

Step 1.5.5.3.11

Factor $- 1$ out of $- 3 y^{2}$ .

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

Step 1.5.5.3.12

Factor $- 1$ out of $- (3 x^{2} - 50) - (3 y^{2})$ .

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

Step 1.5.5.3.13

Simplify the expression.

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Step 1.5.5.3.13.1

Rewrite $- (3 x^{2} - 50 + 3 y^{2})$ as $- 1 (3 x^{2} - 50 + 3 y^{2})$ .

$y' = \frac{x (- 1 (3 x^{2} - 50 + 3 y^{2}))}{y (3 x^{2} + 3 y^{2} + 50)}$

Step 1.5.5.3.13.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 4$ and $y = 2$ .

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Step 1.7.1

Replace the variable $x$ with $4$ in the expression.

$- \frac{(4) (3 {(4)}^{2} - 50 + 3 y^{2})}{y (3 {(4)}^{2} + 3 y^{2} + 50)}$

Step 1.7.2

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

$- \frac{(4) (3 {(4)}^{2} - 50 + 3 {(2)}^{2})}{(2) (3 {(4)}^{2} + 3 {(2)}^{2} + 50)}$

Step 1.7.3

Cancel the common factor of $4$ and $2$ .

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Step 1.7.3.1

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

$- \frac{2 ((2) (3 {(4)}^{2} - 50 + 3 {(2)}^{2}))}{(2) (3 {(4)}^{2} + 3 {(2)}^{2} + 50)}$

Step 1.7.3.2

Cancel the common factors.

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Step 1.7.3.2.1

Cancel the common factor.

$- \frac{2 ((2) (3 {(4)}^{2} - 50 + 3 {(2)}^{2}))}{2 (3 {(4)}^{2} + 3 {(2)}^{2} + 50)}$

Step 1.7.3.2.2

Rewrite the expression.

$- \frac{(2) (3 {(4)}^{2} - 50 + 3 {(2)}^{2})}{3 {(4)}^{2} + 3 {(2)}^{2} + 50}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Raise $4$ to the power of $2$ .

$- \frac{2 (3 \cdot 16 - 50 + 3 \cdot 2^{2})}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.4.2

Multiply $3$ by $16$ .

$- \frac{2 (48 - 50 + 3 \cdot 2^{2})}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.4.3

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

$- \frac{2 (48 - 50 + 3 \cdot 4)}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.4.4

Multiply $3$ by $4$ .

$- \frac{2 (48 - 50 + 12)}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.4.5

Subtract $50$ from $48$ .

$- \frac{2 (- 2 + 12)}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.4.6

Add $- 2$ and $12$ .

$- \frac{2 \cdot 10}{3 \cdot 4^{2} + 3 \cdot 2^{2} + 50}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Raise $4$ to the power of $2$ .

$- \frac{2 \cdot 10}{3 \cdot 16 + 3 \cdot 2^{2} + 50}$

Step 1.7.5.2

Multiply $3$ by $16$ .

$- \frac{2 \cdot 10}{48 + 3 \cdot 2^{2} + 50}$

Step 1.7.5.3

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

$- \frac{2 \cdot 10}{48 + 3 \cdot 4 + 50}$

Step 1.7.5.4

Multiply $3$ by $4$ .

$- \frac{2 \cdot 10}{48 + 12 + 50}$

Step 1.7.5.5

Add $48$ and $12$ .

$- \frac{2 \cdot 10}{60 + 50}$

Step 1.7.5.6

Add $60$ and $50$ .

$- \frac{2 \cdot 10}{110}$

Step 1.7.6

Reduce the expression by cancelling the common factors.

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Step 1.7.6.1

Multiply $2$ by $10$ .

$- \frac{20}{110}$

Step 1.7.6.2

Cancel the common factor of $20$ and $110$ .

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Step 1.7.6.2.1

Factor $10$ out of $20$ .

$- \frac{10 (2)}{110}$

Step 1.7.6.2.2

Cancel the common factors.

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Step 1.7.6.2.2.1

Factor $10$ out of $110$ .

$- \frac{10 \cdot 2}{10 \cdot 11}$

Step 1.7.6.2.2.2

Cancel the common factor.

$- \frac{10 \cdot 2}{10 \cdot 11}$

Step 1.7.6.2.2.3

Rewrite the expression.

$- \frac{2}{11}$

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 $- \frac{2}{11}$ and a given point $(4, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{2}{11} \cdot (x - (4))$

Step 2.2

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

$y - 2 = - \frac{2}{11} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{2}{11} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{2}{11} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = - \frac{2}{11} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = - \frac{2}{11} x - \frac{2}{11} \cdot - 4$

Step 2.3.1.4

Combine $x$ and $\frac{2}{11}$ .

$y - 2 = - \frac{x \cdot 2}{11} - \frac{2}{11} \cdot - 4$

Step 2.3.1.5

Multiply $- \frac{2}{11} \cdot - 4$ .

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Step 2.3.1.5.1

Multiply $- 4$ by $- 1$ .

$y - 2 = - \frac{x \cdot 2}{11} + 4 (\frac{2}{11})$

Step 2.3.1.5.2

Combine $4$ and $\frac{2}{11}$ .

$y - 2 = - \frac{x \cdot 2}{11} + \frac{4 \cdot 2}{11}$

Step 2.3.1.5.3

Multiply $4$ by $2$ .

$y - 2 = - \frac{x \cdot 2}{11} + \frac{8}{11}$

Step 2.3.1.6

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

$y - 2 = - \frac{2 x}{11} + \frac{8}{11}$

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.

$y = - \frac{2 x}{11} + \frac{8}{11} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = - \frac{2 x}{11} + \frac{8}{11} + 2 \cdot \frac{11}{11}$

Step 2.3.2.3

Combine $2$ and $\frac{11}{11}$ .

$y = - \frac{2 x}{11} + \frac{8}{11} + \frac{2 \cdot 11}{11}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{2 x}{11} + \frac{8 + 2 \cdot 11}{11}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $11$ .

$y = - \frac{2 x}{11} + \frac{8 + 22}{11}$

Step 2.3.2.5.2

Add $8$ and $22$ .

$y = - \frac{2 x}{11} + \frac{30}{11}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{2}{11} x) + \frac{30}{11}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{2}{11} x + \frac{30}{11}$

Step 3