One Rule Handles Every Exponent — Until the x Moves Upstairs

You'll learnto run the power rule on any real exponent, to rewrite radicals and reciprocals into power form before you start, and to recognise the one arrangement of symbols the rule says nothing about.

Two moves: the exponent comes down in front, and what is left drops by one. That is the entire rule, and it is not a rule from nowhere — expand the difference quotient for x³ and watch 3x² fall out of it. The rule then covers far more than it looks like it does, because √x and 1/x³ and x√x are all powers once you write them that way. What it does not cover is a variable sitting in the exponent. 2ˣ is not a power of x, and running the rule on it claims a slope of 12 where the curve is climbing at 5.55.

8 STEPS · 6 QUICK CHECKS · REWRITE FIRST, THEN ONE RULE · x IN THE BASE, NEVER IN THE EXPONENT · v1

d/dx[ xⁿ ] = n xⁿ⁻¹ · every real n

Before you start

What you're looking at

A single power, x⁵, with the two moves of the power rule annotated on it — the exponent coming down in front, and what is left dropping by one.

The question

Where does that rule come from, how far does it reach once radicals and reciprocals are rewritten as powers, and where exactly does it stop?

Watch for

The chord on x³ settling on 3x², the rewrite table, √x drawn above its own derivative, and the last step where the rule is run on 2ˣ and the tangent it predicts cuts straight through the curve.

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