Calculus Examples

$(x^{2} + 4) y = 8$ , $(2, 1)$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} ((x^{2} + 4) y) = \frac{d}{d x} (8)$

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 $f (x) = x^{2} + 4$ and $g (x) = y$ .

$(x^{2} + 4) \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2} + 4]$

Step 1.2.2

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

$(x^{2} + 4) y' + y \frac{d}{d x} [x^{2} + 4]$

Step 1.2.3

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$(x^{2} + 4) y' + y (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])$

Step 1.2.4

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

$(x^{2} + 4) y' + y (2 x + \frac{d}{d x} [4])$

Step 1.2.5

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

$(x^{2} + 4) y' + y (2 x + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.2.6.2

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

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

Step 1.2.7

Simplify.

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Step 1.2.7.1

Apply the distributive property.

$x^{2} y' + 4 y' + 2 y x$

Step 1.2.7.2

Reorder terms.

$x^{2} y' + 2 x y + 4 y'$

Step 1.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

$y' x^{2} + y' \cdot 4 = - 2 x y$

Step 1.5.2.3

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

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $x^{2} + 4$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Simplify the denominator.

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Step 1.7.3.1

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

$- 2 \cdot 2 \cdot \frac{1}{4 + 4}$

Step 1.7.3.2

Add $4$ and $4$ .

$- 2 \cdot 2 \cdot \frac{1}{8}$

Step 1.7.4

Reduce the expression by cancelling the common factors.

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Step 1.7.4.1

Multiply $- 2$ by $2$ .

$- 4 \cdot \frac{1}{8}$

Step 1.7.4.2

Cancel the common factor of $4$ .

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Step 1.7.4.2.1

Factor $4$ out of $- 4$ .

$4 (- 1) \cdot \frac{1}{8}$

Step 1.7.4.2.2

Factor $4$ out of $8$ .

$4 \cdot - 1 \cdot \frac{1}{4 \cdot 2}$

Step 1.7.4.2.3

Cancel the common factor.

$4 \cdot - 1 \cdot \frac{1}{4 \cdot 2}$

Step 1.7.4.2.4

Rewrite the expression.

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

Step 1.7.4.3

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$- \frac{1}{2}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $x$ and $\frac{1}{2}$ .

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

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.3.1.5.2

Factor $2$ out of $- 2$ .

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

Step 2.3.1.5.3

Cancel the common factor.

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

Step 2.3.1.5.4

Rewrite the expression.

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

Step 2.3.1.6

Multiply $- 1$ by $- 1$ .

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

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.

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

Step 2.3.2.2

Add $1$ and $1$ .

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

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

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

Step 2.3.3.2

Remove parentheses.

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

Step 3