6.1k views
4 votes
a) Suppose π(Q)=QP(Q)−cQ, where P is a differentiable function and c is a constant. Find an expression for dπ/dQ. b) Suppose π(L)=PF(L)−wL, where F is a differentiable function and P and w are constants. Find an expression for dπ/dL.

User Vittoria
by
8.7k points

1 Answer

6 votes

Final answer:

To find an expression for dπ/dQ, we use the product rule to differentiate the function π(Q). Similarly, to find an expression for dπ/dL, we use the product rule to differentiate the function π(L).

Step-by-step explanation:

To find an expression for dπ/dQ, we need to take the derivative of the function π(Q) with respect to Q. Given that π(Q) = QP(Q) - cQ, where P is a differentiable function and c is a constant, let's find the derivative.

Using the product rule, we differentiate the first term QP(Q) and the second term -cQ separately. The derivative of the first term is Q * dP/dQ + P(Q) and the derivative of the second term is -c.

Combining the derivatives, we get dπ/dQ = Q * dP/dQ + P(Q) - c.

Similarly, for part b), to find an expression for dπ/dL, we need to take the derivative of the function π(L) with respect to L. Given that π(L) = PF(L) - wL, where F is a differentiable function and P and w are constants, we can differentiate the first term PF(L) and the second term -wL separately using the product rule.

The derivative of the first term is P * dF/dL + F(L) and the derivative of the second term is -w.

Combining the derivatives, we get dπ/dL = P * dF/dL + F(L) - w.

User N Alex
by
8.1k points

Related questions

1 answer
5 votes
221k views
1 answer
2 votes
8.3k views