Calculus Examples

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

Step 1

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

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Step 1.1

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 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply.

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Step 1.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.3.2

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

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

Step 1.2.4

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 1.2.5

Simplify the expression.

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Step 1.2.5.1

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

$x^{- 2} + 0$

Step 1.2.5.2

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

$x^{- 2}$

Step 1.3

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

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

Step 1.4

Evaluate the derivative at $x = 1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$\frac{1}{1}$

Step 1.5.2

Divide $1$ 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 $(1, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x - 1$ by $1$ .

$y + 1 = x - 1$

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

Subtract $1$ from both sides of the equation.

$y = x - 1 - 1$

Step 2.3.2.2

Subtract $1$ from $- 1$ .

$y = x - 2$

Step 3