Calculus Examples

$f (x) = x + \frac{1}{x}$ , $a = 1$

Step 1

Consider the function used to find the linearization at $a$ .

$L (x) = f (a) + f' (a) (x - a)$

Step 2

Substitute the value of $a = 1$ into the linearization function.

$L (x) = f (1) + f' (1) (x - 1)$

Step 3

Evaluate $f (1)$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = (1) + \frac{1}{1}$

Step 3.2

Simplify $(1) + \frac{1}{1}$ .

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Step 3.2.1

Remove parentheses.

$(1) + \frac{1}{1}$

Step 3.2.2

Divide $1$ by $1$ .

$1 + 1$

Step 3.2.3

Add $1$ and $1$ .

$2$

Step 4

Find the derivative and evaluate it at $1$ .

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Step 4.1

Find the derivative of $f (x) = x + \frac{1}{x}$ .

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 4.1.1.2

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

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

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

$1 - x^{- 2}$

Step 4.1.3

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

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

Step 4.1.4

Reorder terms.

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

Step 4.2

Replace the variable $x$ with $1$ in the expression.

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

One to any power is one.

$- \frac{1}{1} + 1$

Step 4.3.1.2

Cancel the common factor of $1$ .

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Step 4.3.1.2.1

Cancel the common factor.

$- \frac{1}{1} + 1$

Step 4.3.1.2.2

Rewrite the expression.

$- 1 \cdot 1 + 1$

Step 4.3.1.3

Multiply $- 1$ by $1$ .

$- 1 + 1$

Step 4.3.2

Add $- 1$ and $1$ .

$0$

Step 5

Substitute the components into the linearization function in order to find the linearization at $a$ .

$L (x) = 2 + 0 (x - 1)$

Step 6

Simplify.

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Step 6.1

Multiply $0$ by $x - 1$ .

$L (x) = 2 + 0$

Step 6.2

Add $2$ and $0$ .

$L (x) = 2$

Step 7