MATH.17:5.3 - An operator whose useful law has another form

Let P=R[x], the real-coefficient polynomials in one variable, and define D:P -> P by differentiating each monomial: D(a*x^n)=n*a*x^(n-1) for n>0, and D(a)=0 for constants. This constructs an operation on functions through their polynomial expressions.

For p=x^2 and q=x+1:

D(p*q)=3*x^2+2*x.

The product of derivatives is D(p)*D(q)=2*x. The applicable law is instead:

D(p*q)=D(p)*q+p*D(q),

which gives the required result. To establish the law generally, first expand two monomials: differentiating a*b*x^(m+n) gives coefficient (m+n)*a*b, the sum of the two product-rule contributions. Distributing over the finite sums proves it for polynomials.

The same method of working is used as in :5.2: construct the operator, identify the law needed for the proposed use, and establish that law. Here it enables transforming a product expression into its derivative.