The power rule from geometry

What you’ll learn

Last lesson you computed d/dt(t²) = 2t and d/dt(t³) = 3t² by grinding through binomial expansions. Lined up, the answers reveal a pattern, the power rule, and the single capability this lesson builds is to derive that rule from geometry and use it (with two linearity rules) to differentiate any polynomial on sight, so you never expand a binomial again.

You will see why d/dt(t²) = 2t by picturing t² as a growing square: nudge the side by dt and you add two strips (t·dt each) plus a tiny corner (dt²) that vanishes in the limit, so the 2 is the two strips and the t is each strip’s length. The cube gives 3t² the same way (three growing slabs). Generalizing, d/dt(t^n) = n · t^(n-1), where n counts the faces that grow and t^(n-1) is each face’s size, and it holds for negative and fractional powers too (1/t -> -1/t², √t -> 1/(2√t)). You will add the constant-multiple and sum rules, then differentiate a polynomial like 3t⁴ + 2t² - 7 in three quick lines.

Where this fits

This is lesson 3 of Phase 1 (What a derivative is). Lesson 2 defined the derivative as a limit and computed a couple the slow way; this lesson finds the pattern and the reason behind it, so differentiation becomes fast. The same nudge-and-look method carries straight into lesson 4 (trig derivatives, from a point on the unit circle) and Phase 2’s product and chain rules. The method, not just the rule, is the point: a derivative rule is a consequence of how a quantity grows when you nudge its input.

Before you start

Prerequisite (within this track): lesson 2, The derivative as a rate, since the power rule is the pattern in the t² and t³ derivatives computed there, and “let dt shrink to zero” is the move that drops the corner term. You need comfort with exponent notation (rewriting √t as t^(1/2) and 1/t as t^(-1)) and basic algebra; no prior calculus beyond lesson 2, no coding, nothing installed. The practice is pen and paper.

By the end, you’ll be able to

• Derive the power rule from the geometry of a growing square and a growing cube, rather than from memorization
• Explain why higher-order dt terms (the corner and edge pieces) vanish while the strips and slabs survive
• Apply the power rule to negative and fractional exponents
• Use the constant-multiple and sum rules to differentiate any polynomial term by term

Time and difficulty

• Read time: about 11 minutes
• Practice time: about 13 minutes (differentiating polynomials and fractional/negative powers, a numeric growing-square check, and flashcards)
• Difficulty: standard