Calculus Examples

$f (x) = 6 x + 2$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

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

$6 x + 6 h + 2$

Step 2.2

Reorder $6 x$ and $6 h$ .

$6 h + 6 x + 2$

Step 2.3

Find the components of the definition.

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

$f (x) = 6 x + 2$

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

$f (x) = 6 x + 2$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $2$ out of $6 x + 2$ .

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Step 4.1.1.1

Factor $2$ out of $6 x$ .

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

Step 4.1.1.2

Factor $2$ out of $2$ .

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

Step 4.1.1.3

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

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

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

Step 4.1.2

Multiply $- 1$ by $2$ .

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

Step 4.1.3

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

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Step 4.1.3.1

Factor $2$ out of $6 h$ .

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

Step 4.1.3.2

Factor $2$ out of $6 x$ .

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

Step 4.1.3.3

Factor $2$ out of $2$ .

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

Step 4.1.3.4

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

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

Step 4.1.3.5

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

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

Step 4.1.3.6

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

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

Step 4.1.3.7

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

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

Step 4.1.4

Apply the distributive property.

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

Step 4.1.5

Multiply $3$ by $- 1$ .

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

Step 4.1.6

Multiply $- 1$ by $1$ .

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

Step 4.1.7

Subtract $3 x$ from $3 x$ .

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

Step 4.1.8

Add $3 h$ and $0$ .

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

Step 4.1.9

Subtract $1$ from $1$ .

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

Step 4.1.10

Add $3 h$ and $0$ .

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

Step 4.1.11

Multiply $2$ by $3$ .

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

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $6$ by $1$ .

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

Step 5

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

$6$

Step 6

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