Calculus Examples

$f (x) = 2 x - 3$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

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

$2 x + 2 h - 3$

Step 2.2

Reorder $2 x$ and $2 h$ .

$2 h + 2 x - 3$

Step 2.3

Find the components of the definition.

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

$f (x) = 2 x - 3$

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

$f (x) = 2 x - 3$

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

Multiply $- 1$ by $- 3$ .

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

Step 4.1.4

Subtract $2 x$ from $2 x$ .

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

Step 4.1.5

Add $2 h$ and $0$ .

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

Step 4.1.6

Add $- 3$ and $3$ .

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

Step 4.1.7

Add $2 h$ 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