Symbolic Differentiation

We here motivate the basic rules used in symbolic computation of derivatives as the essence of Symbolic Calculus. Using these rules derivates of endless combinations  (algebraic and composite) of functions with known derivatives, can be computed in a veritable eldorado for lovers of symbolic computation.

Let F(t) and G(t) be (real-valued) functions of time t and let D = d/dt denote differentiation with respect to time, so that dF = DF*dt (or dF(t) = DF(t)*dt) with formally dF(t) = F(t+dt) – F(t) up to second order in dt.

Derivative of a Polynomial: dt^n/dt = n*t^(n-1)

The basic application of symbolic differentiation concerns the symbolic computation of the derivative D = d/dt of a polynomial

  • p(t) = a0 + a1*t + a2*t^2 + a3*t^3 + … + an*t^n

with t^2 = t*t,  t^3 = t*t*t and t^n = t*t*t…*t (n times). The basic case is p(t) = t, with

  • dp(t) = p(t+dt) – p(t) = t+dt – t = dt = 1*dt

with thus dp/dt = dt/dt = 1. Next, with p(t) = t*t we have

  • dp(t) = (t+dt)*(t+dt) – t*t = t*t + 2*t*dt + dt*dt – t*t = 2*t*dt + dt*dt

with thus dp/dt = d(t*t)/dt = 2*dt up to a term scaling with dt. Next, with p(t)=t^3 with have

  • dp(t) = (t+dt)^3 – t^3 = t^3 + 3*t^2*dt + 3*t*dt^2 + dt*3 – t^3 = 3*t^2*dt + 3*t*dt^2 + dt^3

with thus dt^3 = 3*t^2 up to terms scaling with dt or dt^2. More generally, we have

  • dt^n = n*t^(n-1) for n = 1, 2, 3,….

Computation of derivatives of polynomials (in several variables) is a basic ingredient of FEM.

Compare with p5.js code polynomials.

Derivative of a Linear Combination

We have with a and b constants

  • D(a*F + b*G) = a*DF + b*DG,


  • d(a*F + b*G)
  • = (a*F + b*G)(t+dt) -(a*F + b*G)(t) = a*(F(t+dt)-F(t)) + b*(G(t+dt)-G(t))
  • = a*dF + b*dG.

Derivative of Product

We have

  • D(F*G) = DF*G + F*DG,


  • d(F*G) – dF*G – F*dG = F(t+dt)*G(t+dt) – F(t)G(t) – dF*G(t) – F(t)*dG
  • = F(t+dt)*(G(t+dt)-G(t)) – F(t)*dG + G(t)*(F(t+dt) – F(t)) – G(t)*dF
  • = 0

up to send order i dt.

Derivative of a Quotient

We have

  • D(1/G) = – DG/G^2     (with G^2 = G*G)


  • d(1/G) = 1/(G(t+dt) – 1/G(t) = – (G(t+dt) – G(t))/(G(t+dt)*G(t))
  • = – dG/G(t)^2

up second order in dt. More generally

  • D(F/G) = (DF*G-F*DG)/G^2   if G not zero.

The Chain Rule

For the composite function H(t)=F(G(t)) we have

  • DH(t) = DF(G(t))*DG(t)      (with DF(s) = dF(s)/ds and s = G(t))


  • dH = H(t+dt)-H(t)=F(G(t+dt)-F(G(t)) = DF(G(t))*(G(t+dt) – G(t))
  • =DF(G(t))*DG(t)*dt

up to terms of order in dt.

To Think About

  • How to compute the derivate of a function if symbolic differentiation is not an option, because it is too difficult, or the function is not given by an analytic expression.


Leibniz’s first paper on calculus, Acta Eruditorum, 1684, with the above rules for differentiation.

  • The ultimate reason of things must lie in a necessary substance, in which the differentiation of thechanges only exists eminently as in their source; and this is what we call God. (Leibniz)
  • The freedom of thought is a sacred right of every individual man, and diversity will continue to increase with the progress, refinement, and differentiation of the human intellect. (Felix Adler)
  • How did Biot arrive at the partial differential equation? [the heat conduction equation] . . .Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals. (Clifford Truesdell)
  • Common integration is only the memory of differentiation. (Augustus De Morgan)


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