How do solve for q for the following and find the derivative for them?
1) pq+p+100q=50000
2) (p+1)(�ã q+1)=1000
3) p=(1/2)ln((5000-q)/(q+1))
To solve for q in each equation, we will use algebraic manipulation. After finding q, we can then find the derivative of each equation with respect to q.
1) pq+p+100q=50000:
To solve for q, we need to isolate q on one side of the equation. Here's how you can do it:
Step 1: Move all terms involving q to one side by subtracting p from both sides:
pq + 100q = 50000 - p
Step 2: Factor out q from the left side of the equation:
q(p + 100) = 50000 - p
Step 3: Divide both sides by (p + 100) to solve for q:
q = (50000 - p) / (p + 100)
To find the derivative, we need to differentiate both sides of the equation with respect to q. The derivative of q with respect to q is simply 1. The derivative of p with respect to q is 0 since p does not involve q. The derivative of the constant term -p / (p + 100) with respect to q is 0 since it is a constant. Therefore, the derivative of the equation with respect to q is 1.
2) (p+1)(�ã q+1)=1000:
To solve for q, we need to isolate q on one side of the equation. Here's how you can do it:
Step 1: Divide both sides by (p + 1):
�ã q+1 = 1000 / (p + 1)
Step 2: Subtract 1 from both sides:
�ã q = 1000 / (p + 1) - 1
To find the derivative, we need to differentiate both sides of the equation with respect to q. The derivative of �ã q with respect to q is 1. The derivative of 1000 / (p + 1) - 1 with respect to q is 0 since it is a constant. Therefore, the derivative of the equation with respect to q is 1.
3) p = (1/2)ln((5000 - q) / (q + 1)):
To solve for q, we need to isolate q on one side of the equation. Here's how you can do it:
Step 1: Multiply both sides by 2 to get rid of the fraction:
2p = ln((5000 - q) / (q + 1))
Step 2: Convert the equation into exponential form:
(5000 - q) / (q + 1) = e^(2p)
Step 3: Cross-multiply to get rid of the fraction:
(5000 - q) = e^(2p) * (q + 1)
Step 4: Distribute e^(2p) on the right side:
5000 - q = e^(2p)q + e^(2p)
Step 5: Move all terms involving q to one side by subtracting e^(2p)q from both sides:
5000 - e^(2p)q = e^(2p) + q
Step 6: Solve for q by moving all q terms to one side:
5000 - e^(2p) - q = e^(2p)q
Step 7: Combine like terms:
5000 - e^(2p) = e^(2p)q + q
Step 8: Factor out q on the right side:
5000 - e^(2p) = q(e^(2p) + 1)
Step 9: Divide both sides by (e^(2p) + 1) to solve for q:
q = (5000 - e^(2p)) / (e^(2p) + 1)
To find the derivative, we need to differentiate both sides of the equation with respect to q. The derivative of q with respect to q is 1. The derivative of (5000 - e^(2p)) / (e^(2p) + 1) with respect to q can be found using the quotient rule. The derivative of 5000 - e^(2p) with respect to q is 0 since it does not contain q. The derivative of e^(2p) + 1 with respect to q is also 0 since it is a constant. Therefore, the derivative of the equation with respect to q is 1.