Calculus Examples

$f (x) = \frac{1}{x - 5}$ , $(4, - 1)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = - 1$ to find the slope of the tangent line.

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Step 1.1

Rewrite $\frac{1}{x - 5}$ as ${(x - 5)}^{- 1}$ .

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

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x - 5$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x - 5]$

Step 1.2.2

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

$- u^{- 2} \frac{d}{d x} [x - 5]$

Step 1.2.3

Replace all occurrences of $u$ with $x - 5$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

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

Step 1.3.4.2

Multiply $- 1$ by $1$ .

$- {(x - 5)}^{- 2}$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.5

Evaluate the derivative at $x = 4$ .

$- \frac{1}{{((4) - 5)}^{2}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the denominator.

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Step 1.6.1.1

Subtract $5$ from $4$ .

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

Step 1.6.1.2

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

$- \frac{1}{1}$

Step 1.6.2

Reduce the expression by cancelling the common factors.

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Step 1.6.2.1

Cancel the common factor of $1$ .

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Step 1.6.2.1.1

Cancel the common factor.

$- \frac{1}{1}$

Step 1.6.2.1.2

Rewrite the expression.

$- 1 \cdot 1$

Step 1.6.2.2

Multiply $- 1$ by $1$ .

$- 1$

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

$y - (- 1) = - 1 \cdot (x - (4))$

Step 2.2

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

$y + 1 = - 1 \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 1 \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y + 1 = 0 + 0 - 1 \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 1 = - 1 \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y + 1 = - 1 x - 1 \cdot - 4$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

Rewrite $- 1 x$ as $- x$ .

$y + 1 = - x - 1 \cdot - 4$

Step 2.3.1.4.2

Multiply $- 1$ by $- 4$ .

$y + 1 = - x + 4$

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 $1$ from both sides of the equation.

$y = - x + 4 - 1$

Step 2.3.2.2

Subtract $1$ from $4$ .

$y = - x + 3$

Step 3