Calculus Examples

$f (x) = 3 x + 2$

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.

$f (x + h) = 3 (x + h) + 2$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 3 x + 3 h + 2$

Step 2.1.2.2

The final answer is $3 x + 3 h + 2$ .

$3 x + 3 h + 2$

Step 2.2

Reorder $3 x$ and $3 h$ .

$3 h + 3 x + 2$

Step 2.3

Find the components of the definition.

$f (x + h) = 3 h + 3 x + 2$

$f (x) = 3 x + 2$

$f (x + h) = 3 h + 3 x + 2$

$f (x) = 3 x + 2$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{3 h + 3 x + 2 - (3 x + 2)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{3 h + 3 x + 2 - (3 x) - 1 \cdot 2}{h}$

Step 4.1.2

Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{3 h + 3 x + 2 - 3 x - 1 \cdot 2}{h}$

Step 4.1.3

Multiply $- 1$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{3 h + 3 x + 2 - 3 x - 2}{h}$

Step 4.1.4

Subtract $3 x$ from $3 x$ .

$f' (x) = lim_{h \to 0} \frac{3 h + 0 + 2 - 2}{h}$

Step 4.1.5

Add $3 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h + 2 - 2}{h}$

Step 4.1.6

Subtract $2$ from $2$ .

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

Step 4.1.7

Add $3 h$ and $0$ .

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

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $3$ by $1$ .

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

Step 5

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$3$

Step 6