if the equation is P^2-Q=4 find DP/DQ?

Question

The equation given is p^2-Q=4 and I need to solve for DP/DQ. Can anyone help?

Answer 1

Certainly

This is a great instance where implicit differentiation can be used.

Differentiate every term with respect to p:

d[P^2]/dQ - d[Q]/dQ = d[4]/dQ

2P * dP/dQ - 1 = 0

dP/dQ * 2P = 1

dP/dQ = 1 / (2P)



Often with problems involving implicit differentiation it is appropriate to leave it as a function of both the dependent and the independent variables (in this case ONLY the dependent variable) this is in contrast to ordinary differentiation where the derivative is expressed solely a function of the independent variable. However, if the question were posed as "find dP/dQ as a function of Q" then you would take the following steps:



(from above)

dP/dQ = 1 / (2P) and P^2 - Q = 4 so P = +/- sqrt(4+Q)

and dP/dQ = +/- 1 / 2sqrt(4+Q) where +/- reads "plus or minus"



You'll find that if you want to isolate P from the start and proceed without implicit differentiation then you will receive the same answer (after applying the quotient rule). I can tell from the problem that it's a job for implicit differentiation, and I recommend you embrace it as it is a very powerful technique later on in calculus and physics.



Best Regards!

Answer 2

because of the fact the p is squared you initiate with 2 brackets that seem as though this (p )(p ) next you decide on 2 numbers that upload at the same time to make 7 and multiply at the same time to make 12. the numbers that multiply at the same time to make 12 are (a million,12) (2,6) (3,4) now come to a style which of those upload at the same time to furnish you 7 (3+4=7) so which you finished the brackets with the three and the 4 (p+3)(p+4)=0. you are able to now see that in case you multiply 2 values at the same time to =0 then the two a form of values must be 0. i.e. 6*0=0 or 0*4=0, so all of us understand that the two of the brackets =0. If p+3=0 then p=-3 and if p+4=0 then p=-4 so which you will possibly be able to desire to sparkling up the equation the answer is p=-3 or p=-4

Answer 3

q = p^2 - 4



dq/dp = 2p



dp/dq = 1/(dq/dp) which will be an expression in terms of p.



If you need it in terms of q, then solve for p in terms of q and substitute.

Answer 4

p^2 - q = 4

p^2 = q + 4

p = +/- sqrt(q + 4)

dp/dq = +/- 1 / (2 * sqrt(q + 4))