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Equation

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

Examples

Function: $x^2$

Step 1: Find $f(x+h)$

f(x+h) = (x+h)^2 = x^2 + 2xh + h^2

Step 2: Subtract $f(x)$

f(x+h) - f(x) = 2xh + h^2

Step 3: Divide by $h$

\frac{2xh + h^2}{h} = 2x + h

Step 4: Take the limit

f'(x) = 2x

Function: $x^3$

Step 1: Find $f(x+h)$

f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Step 2: Subtract $f(x)$

f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3

Step 3: Divide by $h$

\frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2

Step 4: Take the limit

f'(x) = 3x^2