I need to find dq2/dpb and dqb/dpa
qa=2500+(600/pb+2)-40pa
qb=3000-100pb+(400/pa+5)
a and b are subscripts.
To find dq2/dpb, we need to differentiate the equation qb with respect to pb. Similarly, to find dqb/dpa, we need to differentiate the equation qa with respect to pa.
Let's start with finding dq2/dpb.
Given: qb = 3000 - 100pb + (400/pa + 5)
To differentiate qb with respect to pb, we treat pb as the variable and everything else as a constant. Let's go step by step:
1. Differentiate the first term: The derivative of 3000 with respect to pb is 0 because it does not depend on pb.
So, the derivative of 3000 is 0.
2. Differentiate the second term: The derivative of -100pb with respect to pb is -100.
So, the derivative of -100pb is -100.
3. Differentiate the third term: Since pb is in the denominator, we need to use the chain rule.
The derivative of (400/pa + 5) with respect to pb is (400/pa + 5) * (-400/pb^2)
So, the derivative of (400/pa + 5) is (400/pa + 5) * (-400/pb^2).
4. Combine the derivatives: Add up the derivatives from steps 1, 2, and 3.
dq2/dpb = 0 - 100 + (400/pa + 5) * (-400/pb^2)
Therefore, dq2/dpb = -100 - (400/pa + 5) * (400/pb^2)
Now let's find dqb/dpa.
Given: qa = 2500 + (600/pb + 2) - 40pa
To differentiate qa with respect to pa, we treat pa as the variable and everything else as a constant. Let's go step by step:
1. Differentiate the first term: The derivative of 2500 with respect to pa is 0 because it does not depend on pa.
So, the derivative of 2500 is 0.
2. Differentiate the second term: Since pa is in the denominator, we need to use the chain rule.
The derivative of (600/pb + 2) with respect to pa is (600/pb + 2) * (-600/pa^2)
So, the derivative of (600/pb + 2) is (600/pb + 2) * (-600/pa^2).
3. Differentiate the third term: The derivative of -40pa with respect to pa is -40.
So, the derivative of -40pa is -40.
4. Combine the derivatives: Add up the derivatives from steps 1, 2, and 3.
dqb/dpa = 0 + (600/pb + 2) * (-600/pa^2) - 40
Therefore, dqb/dpa = (600/pb + 2) * (-600/pa^2) - 40