True or False

One algebraic claim on the board — say, the derivative of a product is the product of the derivatives — and one job: put it on trial. Is it ever true? Always true? An equation is not a fact just because it is written down; it is a claim about functions now, not just numbers, and claims earn their verdicts in court. The stakes are real: this particular claim feels so natural that some part of every class believes it, quietly, until the trial.

How we play

1. Vote first — true, false, or “it depends” — before any working.
2. Prosecute and defend: test functions, rewrite, sketch, whatever bites.
3. Deliver a verdict with evidence: always, never, or exactly when.

The trial of $(fg{)}^{\prime }=f^{\prime }\cdot g^{\prime }$

• “Try $f(x)=x^2$ and $g(x)=x^3$. The product is $x^5$, whose derivative is $5x^4$. The product of the derivatives is $2x\cdot 3x^2=6x^3$. At $x=0$ both come out to 0 — looks true.”
• “Try $x=1$: the left side gives 5, the right side gives 6. False.”
• “Graph both: $5x^4$ and $6x^3$ are different shapes entirely — they cross at exactly two points and disagree everywhere else.”
• Verdict: false — true only at scattered points, never as a rule. And the reason is worth the trial: when $f$ and $g$ both grow, the product grows for two reasons at once — each factor’s growth gets scaled by the other factor’s size. The honest relation is $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$, each term one of the two reasons, and it comes from Derivative Rules. Check it on the evidence: $2x\cdot x^3+x^2\cdot 3x^2=5x^4$. Exactly right.

One variation

Claims that sound like the same trap but are not: “the derivative of a sum is the sum of the derivatives.” Every test passes — and this one is always true, for any two differentiable functions. Sums split and products do not, and saying why the two claims meet different fates is worth more than either verdict. One more for the docket: “the derivative of $\sin x$ is $\cos x$.” Always — but only because this course measures angles in radians, a fine-print clause Derivatives of Sinusoidal Functions reads aloud.

One witness is not a proof

A single counter-example kills an “always” — $x=1$ ended the trial above. A single confirming example proves nothing: $x=0$ testified for the claim and the claim was still false. To speak about all functions you need a reason, not a coincidence — and testing a value you were not given is Checking Your Own Work wearing a courtroom robe.