Basic Derivative Rules from First Principles

Calculus 1 · Axiom Academy

From First Principles — watch the constant, power, constant-multiple, and sum rules each fall straight out of the limit definition. Every rule below starts here. The derivative is the limit of the difference quotient — the slope of the secant through (x, f(x)) and (x+h, f(x+h)) as the gap h shrinks to zero. Take f(x) = c , a flat horizontal line. Feed it into the definition: the numerator is f(x+h) - f(x) = c - c , which is just 0 . The big one: the derivative of x^n . Expand (x+h)^n with the binomial theorem and watch the pieces sort themselves — only one survives the limit. The leading x^n cancels with the -x^n ; divide every remaining term by h ; then let and every term that still carries an h vanishes. Scaling a function by a constant k scales its difference quotient by the same k — so the constant just slides out in front of the limit, untouched. 5. The Sum and Difference Rules For f(x) + g(x) , the single difference quotient splits cleanly into one quotient for f plus one for g . The limit of a sum is the sum of the limits, so the derivatives simply add. You've watched all four basic derivative rules fall out of one limit definition — and you can now differentiate any polynomial by inspection. Scroll up to replay any derivation.

This is the written version of the interactive lesson above. See the full Calculus 1 course.