Mathematics

The derivative from first principle means using the limit definition $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ rather than memorized rules. The key insight is to substitute $x+h$ into your function, expand carefully, subtract the original function, divide by $h$, and then let $h$ shrink to zero—the $h$ terms in the denominator will cancel, leaving you with a clean expression. This process works for any function and shows why derivative rules work.

The derivative from first principle means using the limit definition $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ rather than memorized rules. The key insight is to substitute $x+h$ into your function, expand carefully, subtract the original function, divide by $h$, and then let $h$ shrink to zero—the $h$ terms in the denominator will cancel, leaving you with a clean expression. This process works for any function and shows why derivative rules work.