Loading...
Loading...
Calculus 1 · Axiom Academy
No limits, no tangent lines — just a table of derivatives and one pattern your eye can catch. One pattern hiding in a table of derivatives Finding a derivative from the definition — limits, tangent lines, all of it — is real work. But for the building blocks of every polynomial, the powers x, x², x³, … , the answers fall into a pattern so clean you can read off the next one without doing any of that work. The whole trick is to see it first. Watch the table fill in one row at a time. In each row the exponent drops down to the front as a coefficient, and the power left behind goes down by one . By the last row you'll be predicting it before it lands. Same move every row: bring the exponent down out front, subtract one from the power that stays. Pick a power. Predict its derivative. Set the exponent n , look at f(x) = x n , and call the answer in your head before you reveal it. Then hit Reveal the derivative and watch the exponent drop to the front and the power step down — the same move you just saw, now for a power you chose. Coefficient = the old exponent. New power = one less. Every single time. Every row of that table is the same statement with a different number. Here it is once, in general — the power rule . Drag n and watch the rule fill its own blanks: the exponent flows into the front as the coefficient, and the power becomes n − 1 . The general law and the concrete row are the same thing. d/dx[xⁿ] = n·x^(n−1) — that one line is the whole pattern, written once.
This is the written version of the interactive lesson above. See the full Calculus 1 course.