Product Rule: (x^a · e^(bx))′

For $f(x) = u(x)v(x)$, $f'(x) = u'(x)v(x) + u(x)v'(x)$. So $(x^2\sin x)' = 2x\sin x + x^2\cos x$. The two terms cannot be combined — each one captures change in one factor while the other stays fixed.

Pick functions $u(x)$ and $v(x)$ via the controls. The lab plots $u\cdot v$ along with the derivative $u'v + uv'$ and reports the slope at a chosen $x_0$. Compare with the slope read directly off the product curve to confirm the formula.

Students sometimes write $(uv)' = u'v'$, which is wrong (that would mean derivative of $x^2 = x\cdot x$ is $1\cdot 1 = 1$, but it's actually $2x$). The product rule requires both terms. The lab plots the wrong-formula and right-formula side by side so the difference shrieks.

Physics: rate of change of momentum $p = mv$ when both mass and velocity change (rocket equation). Economics: revenue $R = pq$ — when price and quantity both move, $dR/dt = (dp/dt)q + p(dq/dt)$. The product rule is the chain that links co-evolving quantities.