2.4.2—Product and quotient rules

For y=uv, y′=u′v+uv′. For y=u/v, y′=(u′v−uv′)/v². Both rules follow from the product rule and reciprocal structure.

Label u and v before differentiating, keep the denominator squared in the quotient rule, and use a logarithmic derivative when products of powers are cumbersome.

d[x²e^x]/dx=e^x(x²+2x); d[(sin x)/x]/dx=(x cosx−sinx)/x².

Differentiating numerator and denominator separately is not the quotient rule.