Calculus Examples

$f (x) = \frac{1}{x}$ ; $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Divide $1$ by $1$ .

$y = 1$

Step 2

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 2.1

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

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

Step 2.2

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

$- x^{- 2}$

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 $x = 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

One to any power is one.

$- \frac{1}{1}$

Step 2.5.2

Cancel the common factor of $1$ .

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Step 2.5.2.1

Cancel the common factor.

$- \frac{1}{1}$

Step 2.5.2.2

Rewrite the expression.

$- 1 \cdot 1$

Step 2.5.3

Multiply $- 1$ by $1$ .

$- 1$

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 $- 1$ 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 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

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

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

Simplify the expression.

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Step 3.3.1.4.1

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

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

Step 3.3.1.4.2

Multiply $- 1$ by $- 1$ .

$y - 1 = - x + 1$

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 $1$ to both sides of the equation.

$y = - x + 1 + 1$

Step 3.3.2.2

Add $1$ and $1$ .

$y = - x + 2$

Step 4