Calculus Examples

$x y = 15$ , $x = 5$

Step 1

Find the corresponding $y$ -value to $x = 5$ .

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Step 1.1

Substitute $5$ in for $x$ .

$(5) \cdot y = 15$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Multiply $5$ by $y$ .

$5 \cdot y = 15$

Step 1.2.2

Divide each term in $5 \cdot y = 15$ by $5$ and simplify.

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Step 1.2.2.1

Divide each term in $5 \cdot y = 15$ by $5$ .

$\frac{5 \cdot y}{5} = \frac{15}{5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{5 \cdot y}{5} = \frac{15}{5}$

Step 1.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{15}{5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $15$ by $5$ .

$y = 3$

Step 2

Find the first derivative and evaluate at $x = 5$ and $y = 3$ to find the slope of the tangent line.

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Step 2.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x y) = \frac{d}{d x} (15)$

Step 2.2

Differentiate the left side of the equation.

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Step 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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

$x y' + y \cdot 1$

Step 2.2.4

Multiply $y$ by $1$ .

$x y' + y$

Step 2.3

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$0$

Step 2.4

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

$x y' + y = 0$

Step 2.5

Solve for $y'$ .

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Step 2.5.1

Subtract $y$ from both sides of the equation.

$x y' = - y$

Step 2.5.2

Divide each term in $x y' = - y$ by $x$ and simplify.

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Step 2.5.2.1

Divide each term in $x y' = - y$ by $x$ .

$\frac{x y'}{x} = \frac{- y}{x}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $x$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{- y}{x}$

Step 2.5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- y}{x}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Move the negative in front of the fraction.

$y' = - \frac{y}{x}$

Step 2.6

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

$\frac{d y}{d x} = - \frac{y}{x}$

Step 2.7

Evaluate at $x = 5$ and $y = 3$ .

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Step 2.7.1

Replace the variable $x$ with $5$ in the expression.

$- \frac{y}{5}$

Step 2.7.2

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

$- \frac{3}{5}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- \frac{3}{5}$ and a given point $(5, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{3}{5} \cdot (x - (5))$

Step 3.2

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

$y - 3 = - \frac{3}{5} \cdot (x - 5)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{3}{5} \cdot (x - 5)$ .

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Step 3.3.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{3}{5} \cdot (x - 5)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$y - 3 = - \frac{3}{5} x - \frac{3}{5} \cdot - 5$

Step 3.3.1.2.2

Combine $x$ and $\frac{3}{5}$ .

$y - 3 = - \frac{x \cdot 3}{5} - \frac{3}{5} \cdot - 5$

Step 3.3.1.2.3

Cancel the common factor of $5$ .

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Step 3.3.1.2.3.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$y - 3 = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot - 5$

Step 3.3.1.2.3.2

Factor $5$ out of $- 5$ .

$y - 3 = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot (5 (- 1))$

Step 3.3.1.2.3.3

Cancel the common factor.

$y - 3 = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot (5 \cdot - 1)$

Step 3.3.1.2.3.4

Rewrite the expression.

$y - 3 = - \frac{x \cdot 3}{5} - 3 \cdot - 1$

Step 3.3.1.2.4

Multiply $- 3$ by $- 1$ .

$y - 3 = - \frac{x \cdot 3}{5} + 3$

Step 3.3.1.3

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

$y - 3 = - \frac{3 x}{5} + 3$

Step 3.3.2

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

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Step 3.3.2.1

Add $3$ to both sides of the equation.

$y = - \frac{3 x}{5} + 3 + 3$

Step 3.3.2.2

Add $3$ and $3$ .

$y = - \frac{3 x}{5} + 6$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{3}{5} x) + 6$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{3}{5} x + 6$

Step 4