The General Power Rule - A Chain Rule Application

Calculus 1 · Axiom Academy

The chain rule applied to powers: differentiate [g(x)]^n without expanding it first. 1. One Extra Factor on Top of the Power Rule When the base is a whole function g(x) , the answer is the ordinary power rule times the derivative of that inside function. Watch the familiar part appear, then the chain rule snap on its extra factor g'(x) . The power-rule part — bring n down, drop the exponent by one The chain-rule part — multiply by g'(x) , the inside's derivative Every general-power-rule derivative is the same three mechanical moves. Watch them happen on (2x+3)^5 : the exponent drops to the front, the power steps down by one, and the inside's derivative arrives as a new factor. Bring the exponent down: the n becomes a leading factor. Step the power down by one: the exponent goes from n to n-1 . Multiply by the inside's derivative: attach the factor g'(x) . 3. The Factor Everyone Forgets The most common mistake is stopping after the power rule and never multiplying by g'(x) . Take . Here g(x)=3x-2 , so g'(x)=3 . Watch the incomplete answer get corrected by the missing factor. Forgetting the would leave 4(3x-2)^3 — off by the constant factor the inside contributes. A negative power: , with g'(x)=2x : A root, rewritten as a power: , with g'(x)=4 : The habit that never fails: name the inside g(x) , find g'(x) , then write . The general power rule is the power rule with one chain-rule factor stapled on — and that factor, g'(x) , is the whole reason it deserves its own name.

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