Calculus Examples

$f (x) = | x + 1 |$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = | (x + h) + 1 |$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = | x + h + 1 |$

Step 2.1.2.2

The final answer is $| x + h + 1 |$ .

$| x + h + 1 |$

Step 2.2

Find the components of the definition.

$f (x + h) = | x + h + 1 |$

$f (x) = | x + 1 |$

$f (x + h) = | x + h + 1 |$

$f (x) = | x + 1 |$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{| x + h + 1 | - (| x + 1 |)}{h}$

Step 4

Multiply $- 1$ by $| x + 1 |$ .

$f' (x) = lim_{h \to 0} \frac{| x + h + 1 | - | x + 1 |}{h}$

Step 5

Set up the limit as a left-sided limit.

$lim_{h \to 0^{-}} \frac{| x + h + 1 | - | x + 1 |}{h}$

Step 6

Evaluate the limits by plugging in the value for the variable.

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Step 6.1

Evaluate the limit of $\frac{| x + h + 1 | - | x + 1 |}{h}$ by plugging in $0$ for $h$ .

$\frac{| x + 0 + 1 | - | x + 1 |}{0}$

Step 6.2

Since $\frac{| x + 0 + 1 | - | x + 1 |}{0}$ is undefined, the limit does not exist.

$Does not exist$

Step 7

Set up the limit as a right-sided limit.

$lim_{h \to 0^{+}} \frac{| x + h + 1 | - | x + 1 |}{h}$

Step 8

Evaluate the limits by plugging in the value for the variable.

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Step 8.1

Evaluate the limit of $\frac{| x + h + 1 | - | x + 1 |}{h}$ by plugging in $0$ for $h$ .

$\frac{| x + 0 + 1 | - | x + 1 |}{0}$

Step 8.2

Since $\frac{| x + 0 + 1 | - | x + 1 |}{0}$ is undefined, the limit does not exist.

$Does not exist$

Step 9

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$

Step 10