3.4.2—Product and quotient rules

For y=uv use u′v+uv′; for y=u/v use (u′v−uv′)/v². The formulas account for both numerator and denominator variation.

Choose factors that keep algebra manageable, and simplify only after applying the rule. A logarithmic derivative can be shorter for a product of powers.

d[x sin x]/dx=sin x+x cos x; d[(x+1)/(x−1)]/dx=−2/(x−1)².

Differentiating top and bottom independently is invalid for a quotient.