Calculus Examples

$f (x) = 6 x + 2$

Consider the limit definition of the derivative.

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

Find the components of the definition.

Tap for more steps...

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

Tap for more steps...

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

$f (x + h) = 6 (x + h) + 2$

Simplify the result.

Tap for more steps...

Apply the distributive property.

$f (x + h) = 6 x + 6 h + 2$

The final answer is $6 x + 6 h + 2$ .

$6 x + 6 h + 2$

Find the components of the definition.

$f (x + h) = 6 x + 6 h + 2$

$f (x) = 6 x + 2$

$f (x + h) = 6 x + 6 h + 2$

$f (x) = 6 x + 2$

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{6 x + 6 h + 2 - (6 x + 2)}{h}$

Simplify.

Tap for more steps...

Simplify the numerator.

Tap for more steps...

Group the last three terms.

$f' (x) = lim_{h \to 0} \frac{6 x + (6 h + (2 - (6 x + 2)))}{h}$

Factor.

$f' (x) = lim_{h \to 0} \frac{6 x + (6 h + (2 - 1 \cdot (2 (3 x + 1))))}{h}$

Multiply $- 1$ by $2$ to get $- 2$ .

$f' (x) = lim_{h \to 0} \frac{6 x + (6 h + (2 - 2 (3 x + 1)))}{h}$

Remove parentheses.

$f' (x) = lim_{h \to 0} \frac{6 x + 6 h + (2 - 2 (3 x + 1))}{h}$

Remove parentheses.

$f' (x) = lim_{h \to 0} \frac{6 x + 6 h + 2 - 2 (3 x + 1)}{h}$

Factor $2$ out of $6 x + 6 h + 2 - 2 (3 x + 1)$ .

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

Apply the distributive property.

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

Multiply $3$ by $- 1$ to get $- 3$ .

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

Multiply $- 1$ by $1$ to get $- 1$ .

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

Remove parentheses.

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

Subtract $3 x$ from $3 x$ to get $0$ .

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

Add $3 h$ and $0$ to get $3 h$ .

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

Subtract $1$ from $1$ to get $0$ .

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

Factor.

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

Multiply $2$ by $3$ to get $6$ .

$f' (x) = lim_{h \to 0} \frac{6 h}{h}$

Reduce the expression by cancelling the common factors.

Tap for more steps...

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{6 h}{h}$

Divide $6$ by $1$ to get $6$ .

$f' (x) = lim_{h \to 0} 6$

Evaluate the limit of $6$ which is constant as $h$ approaches $0$ .

$6$

Enter YOUR Problem

Calculus Examples

Please Rate Your Tutor