Calculus Examples

$f (x) = 3$

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.

$f (x + h) = 3$

Step 2.1.2

The final answer is $3$ .

$3$

Step 2.2

Find the components of the definition.

$f (x + h) = 3$

$f (x) = 3$

$f (x + h) = 3$

$f (x) = 3$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{3 - (3)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{3 - 3}{h}$

Step 4.1.2

Subtract $3$ from $3$ .

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

Step 4.2

Divide $0$ by $h$ .

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

Step 5

Evaluate the limit of $0$ which is constant as $h$ approaches $0$ .

$0$

Step 6