Differentiate TC(Q) = D*S*Q^(-1) + (H/2)*Q with respect to Q. The first term has derivative -D*S*Q^(-2); the second term has derivative H/2. Sum gives dTC/dQ = -DS/Q^2 + H/2^[1].

Where the purchase term went. If we had included D*C in TC(Q), its derivative would be zero (D*C is constant in Q). So omitting it in step 1 has no effect on this derivative. The first-order condition is identical with or without the purchase term, which is why we drop it.

Geometric meaning. Setting the derivative to zero locates a stationary point of TC(Q). The U-shape of TC(Q) (holding line rising linearly, ordering line falling hyperbolically) means the stationary point is a minimum, not a maximum. The step-4 second derivative formally confirms this.

Setting dTC/dQ = 0 in step 3 yields the closed-form Wilson formula. The geometric signature: at Q*, the ordering-cost line (D/Q)S crosses the holding-cost line (Q/2)H. This is the "equal-cost" condition that gives the U-curve its characteristic minimum^[2].