Calculus Examples

$f (x) = 2 x + 2$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

The final answer is

$2 x + 2 h + 2$ .

$2 x + 2 h + 2$

Step 2.2

Reorder

$2 x$ and

$2 h$ .

$2 h + 2 x + 2$

Step 2.3

Find the components of the definition.

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

$f (x) = 2 x + 2$

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

$f (x) = 2 x + 2$

Step 3

Plug in the components.

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

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

$2 x + 2$ .

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Step 4.1.1.1

Factor

$2$ out of

$2 x$ .

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

Step 4.1.1.2

Factor

$2$ out of

$2$ .

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

Step 4.1.1.3

Factor

$2$ out of

$2 (x) + 2 (1)$ .

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

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

Step 4.1.2

Multiply

$- 1$ by

$2$ .

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

Step 4.1.3

Factor

$2$ out of

$2 h + 2 x + 2 - 2 (x + 1)$ .

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Step 4.1.3.1

Factor

$2$ out of

$2 h$ .

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

Step 4.1.3.2

Factor

$2$ out of

$2 x$ .

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

Step 4.1.3.3

Factor

$2$ out of

$2$ .

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

Step 4.1.3.4

Factor

$2$ out of

$- 2 (x + 1)$ .

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

Step 4.1.3.5

Factor

$2$ out of

$2 h + 2 (x)$ .

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

Step 4.1.3.6

Factor

$2$ out of

$2 (h + x) + 2 (1)$ .

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

Step 4.1.3.7

Factor

$2$ out of

$2 (h + x + 1) + 2 (- (x + 1))$ .

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

Step 4.1.4

Apply the distributive property.

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

Step 4.1.5

Multiply

$- 1$ by

$1$ .

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

Step 4.1.6

Subtract

$x$ from

$x$ .

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

Step 4.1.7

Add

$h$ and

$0$ .

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

Step 4.1.8

Subtract

$1$ from

$1$ .

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

Step 4.1.9

Add

$h$ and

$0$ .

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

Step 4.2

Cancel the common factor of

$h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide

$2$ by

$1$ .

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

Step 5

Evaluate the limit of

$2$ which is constant as

$h$ approaches

$0$ .

$2$

Step 6

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