Calculus Examples

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

Step 1

Find the corresponding $y$ -value to $x = 2$ .

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Step 1.1

Substitute $2$ in for $x$ .

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

Step 1.2

Remove parentheses.

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

Step 2

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

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Step 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 $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

$- \frac{d}{d x} [\frac{1}{x}] + \frac{1}{x} \frac{d}{d x} [- 1]$

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply.

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Step 2.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.3.2

Multiply $x^{- 2}$ by $1$ .

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

Step 2.2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{- 2} + \frac{1}{x} \cdot 0$

Step 2.2.5

Simplify the expression.

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Step 2.2.5.1

Multiply $\frac{1}{x}$ by $0$ .

$x^{- 2} + 0$

Step 2.2.5.2

Add $x^{- 2}$ and $0$ .

$x^{- 2}$

Step 2.3

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

$\frac{1}{x^{2}}$

Step 2.4

Evaluate the derivative at $x = 2$ .

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

Step 2.5

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

$\frac{1}{4}$

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

$y - (- \frac{1}{2}) = \frac{1}{4} \cdot (x - (2))$

Step 3.2

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

$y + \frac{1}{2} = \frac{1}{4} \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{1}{4} \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y + \frac{1}{2} = 0 + 0 + \frac{1}{4} \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y + \frac{1}{2} = \frac{1}{4} \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y + \frac{1}{2} = \frac{1}{4} x + \frac{1}{4} \cdot - 2$

Step 3.3.1.4

Combine $\frac{1}{4}$ and $x$ .

$y + \frac{1}{2} = \frac{x}{4} + \frac{1}{4} \cdot - 2$

Step 3.3.1.5

Cancel the common factor of $2$ .

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Step 3.3.1.5.1

Factor $2$ out of $4$ .

$y + \frac{1}{2} = \frac{x}{4} + \frac{1}{2 (2)} \cdot - 2$

Step 3.3.1.5.2

Factor $2$ out of $- 2$ .

$y + \frac{1}{2} = \frac{x}{4} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 3.3.1.5.3

Cancel the common factor.

$y + \frac{1}{2} = \frac{x}{4} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 3.3.1.5.4

Rewrite the expression.

$y + \frac{1}{2} = \frac{x}{4} + \frac{1}{2} \cdot - 1$

Step 3.3.1.6

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

$y + \frac{1}{2} = \frac{x}{4} + \frac{- 1}{2}$

Step 3.3.1.7

Move the negative in front of the fraction.

$y + \frac{1}{2} = \frac{x}{4} - \frac{1}{2}$

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

Subtract $\frac{1}{2}$ from both sides of the equation.

$y = \frac{x}{4} - \frac{1}{2} - \frac{1}{2}$

Step 3.3.2.2

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{- 1 - 1}{2}$

Step 3.3.2.3

Subtract $1$ from $- 1$ .

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

Step 3.3.2.4

Divide $- 2$ by $2$ .

$y = \frac{x}{4} - 1$

Step 3.3.3

Reorder terms.

$y = \frac{1}{4} x - 1$

Step 4