Total differentiation?

Question

The formula

1/R=(1/r1) + (1/r2)

determines the combined resistance R when resistors r1 and r2 are connected in parallel. Use total differentiation to determine the effect of a decreasing r1 to the combined resistance R.(assume that r2 is fixed.)

Answer 1

If r2 is fixed, you wouldn't use total differentiation; you would use partial differentiation. I don't have access to the proper symbol, so I will still use d/d(r1).

1/R = (1/r1) + (1/r2) = (r1 + r2) / (r1*r2) => R = (r1*r2) / (r1 + r2)
d(R)/d(r1) = d[(r1*r2) / (r1 + r2)]/d(r1) = [(r1 + r2)(r2) - (r1*r2)*1] / (r1 + r2)^2 = [r1*r2 + (r2)^2 - r1*r2] / (r1 + r2)^2 = (r2)^2 / (r1 + r2)^2, where r2 is constant. I used the quotient rule for differentiation.

Answer 2

R = r1*r2/(r1+r2)
dR= 1/(r1+r2)^2 [(r1+r2)*r2-r1*r2)]dr1 = -r2^2/(r1+r2)^2 *dr1