A Tip for Older People: Differential Calculus

Say an equation is factored as y = (x+a)(x+b)(x+c) and so forth.

Then dy/dx = the sum of the various factor products, taken n-1 at a time.

This means that if the factored polynomial has n factors, each factor product has n-1 factors.

Example #1:
   y = x2 + 3x + 2 = (x+1)(x+2)
Then     dy/dx = (x+1) + (x+2)
Example #2:
   y = x4 + 10×3 + 35×2 + 50x + 24 = (x+1)(x+2)(x+3)(x+4) … factor the polynomial, if possible

   dy/dx = (x+2)(x+3)(x+4) + (x+1)(x+3)(x+4) + (x+1)(x+2)(x+4) + (x+1)(x+2)(x+3)
     
   d2y/dx2 = 2{(x+3)(x+4) +(x+2)(x+4) + (x+2)(x+3) + (x+1)(x+4) + (x+1)(x+3) + (x+1)(x+2)}

   d3y/dx3 = 6{(x+1) + (x+2) + (x+3) + (x+4)2}