Calculus Examples

$y = \frac{x + 3}{x - 5}$ , $(4, - 7)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = - 7$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 3$ and $g (x) = x - 5$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{(x - 5) (1 + 0) - (x + 3) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

Multiply $x - 5$ by $1$ .

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

Step 1.2.5

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

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

Step 1.2.6

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

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

Step 1.2.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{x - 5 - (x + 3) (1 + 0)}{{(x - 5)}^{2}}$

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Combine the opposite terms in $x - 5 - x - 1 \cdot 3$ .

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Step 1.3.2.1.1

Subtract $x$ from $x$ .

$\frac{0 - 5 - 1 \cdot 3}{{(x - 5)}^{2}}$

Step 1.3.2.1.2

Subtract $5$ from $0$ .

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

Step 1.3.2.2

Multiply $- 1$ by $3$ .

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

Step 1.3.2.3

Subtract $3$ from $- 5$ .

$\frac{- 8}{{(x - 5)}^{2}}$

Step 1.3.3

Move the negative in front of the fraction.

$- \frac{8}{{(x - 5)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 4$ .

$- \frac{8}{{((4) - 5)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Subtract $5$ from $4$ .

$- \frac{8}{{(- 1)}^{2}}$

Step 1.5.1.2

Raise $- 1$ to the power of $2$ .

$- \frac{8}{1}$

Step 1.5.2

Simplify the expression.

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Step 1.5.2.1

Divide $8$ by $1$ .

$- 1 \cdot 8$

Step 1.5.2.2

Multiply $- 1$ by $8$ .

$- 8$

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 $- 8$ and a given point $(4, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 7) = - 8 \cdot (x - (4))$

Step 2.2

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

$y + 7 = - 8 \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 8 \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y + 7 = 0 + 0 - 8 \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 7 = - 8 \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y + 7 = - 8 x - 8 \cdot - 4$

Step 2.3.1.4

Multiply $- 8$ by $- 4$ .

$y + 7 = - 8 x + 32$

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

Subtract $7$ from both sides of the equation.

$y = - 8 x + 32 - 7$

Step 2.3.2.2

Subtract $7$ from $32$ .

$y = - 8 x + 25$

Step 3