Mistake Master · AP Calculus · Unit 3 · Step-Through Animation
The Second Pass Needs Every Rule the First One Did
You'll learnto treat a second derivative as a fresh differentiation problem — to read the structure of f′ before differentiating it, to finish an implicit second derivative by substituting the first one back in, and to catch a wrong f′′ by holding it against the shape of the curve it claims to describe.
A second derivative is not a special object. It is the derivative of the first derivative, and f′ is a function like any other: whatever rules it needs, it needs again. The trouble is that f′ is usually more complicated than f was. Differentiating x²eˣ once produces a sum of two products. Differentiating sin(2x) once leaves the composition still sitting there, inner factor and all. So the second pass often needs more rules than the first — and the characteristic error of this topic is to treat it as needing fewer, to differentiate f′ as though it were a bare power and let the chain, product or implicit rule quietly expire. It does not expire. Every pass through a composition earns its own inner factor, which is why the second derivative of sin(kx) carries k², and the fourth derivative of e³ˣ carries 3⁴.
8 STEPS · 6 QUICK CHECKS · f′′ IS THE DERIVATIVE OF f′, WITH ALL THE RULES · READ f′ AS A NEW PROBLEM · v1
f′′ = (f′)′ ⇄ every rule, a second time
Before you start
What you're looking at
Three panels stacked on one shared x axis: a function, its derivative, and its second derivative. Every curve below the top one is measured off the curve above it by the same differentiation routine, applied again — there is no separate "second derivative" formula anywhere in this file.
The question
If f′ turned out to be a product, or still contains a composition, what does the second pass owe you — and what does the picture look like when that debt goes unpaid?
Watch for
Step 5, where the implicit answer is not finished until the first derivative is substituted back in, and step 7, where a second pass done as a bare power claims a curve bends upward while the drawn curve plainly bends down.
Mistake Master