Expert: Socrates - 12/30/2009

Question
The power P (watts) of an electric circuit is related to the circuit's resistance R (ohms) and current I (amperes) by the equation P=RI^2

a)How is dP/dt related to dR/dt and dI/dt?

b)How is dR/dt related to dI/dt if P is constant?

I am doing a review on this section, and for some reason I am blanking on this. Can you help me? Thanks!

Answer
Think of P , R and I as functions of t and use the rules for finding the derivatives of products and powers of functions .

I will use the prime (') notation for derivatives then switch back to the d/dt form


P = RI^2

P' = (R')(I^2) + (R)(I^2)'   using the product rule

P' = (R')(I^2) + 2RII'  using the power rule


Now switch notation


dP/dt = I^2 dR/dt  + 2RI dI/dt


This answers part a)


For part b)  , if P is constant , remember that the derivative of a constant function is 0 , so

replace  dP/dt by 0 in the answer to a) and get

0 = I^2 dR/dt  + 2RI dI/dt

solve for dR/dt

I^2 dR/dt = - 2RI dI/dt

divide both sides by I^2

dR/dt = (-2R/I) dI/dt

This is the answer to b)


so here are your answers


a) dP/dt = I^2 dR/dt  + 2RI dI/dt

b) dR/dt = (-2R/I) dI/dt