Question

Find the differentiation with respect to Q of (100-Q)Q - 2 sqrt(wrQ) / sqrt(rQ/w).

A) 100 - 2Q sqrt(w/r)
B) 100 - sqrt(w/r)
C) 100 + 2Q sqrt(w/r)
D) 100 + sqrt(w/r)

Answer 1

To differentiate the expression (100-Q)Q - 2 sqrt(wrQ) / sqrt(rQ/w) with respect to Q, we can use the quotient rule and the chain rule. The differentiation involves applying the quotient rule to one term and the product rule to another term.

Step-by-step explanation:

To find the differentiation with respect to Q of the given expression, we can use the quotient rule and the chain rule. The given expression is: (100-Q)Q - 2 sqrt(wrQ) / sqrt(rQ/w). Let's differentiate the expression step by step.

1. Apply the quotient rule to the term - 2 sqrt(wrQ) / sqrt(rQ/w), remembering that sqrt(a) = a^0.5:
   d/dQ (-2 sqrt(wrQ) / sqrt(rQ/w)) = [0.5*(-2)*sqrt(wrQ)*(rQ/w)^(-0.5) - (-2 sqrt(wrQ))*(0.5)*(rQ/w)^(-1.5)] / (rQ/w)
2. Apply the product rule to the term (100-Q)Q:
   d/dQ [(100-Q)Q] = 100Q + (100-Q) - Q

Combining both differentiations, we get:
d/dQ [(100-Q)Q - 2 sqrt(wrQ) / sqrt(rQ/w)] = 100Q + (100-Q) - Q + [0.5*(-2)*sqrt(wrQ)*(rQ/w)^(-0.5) - (-2 sqrt(wrQ))*(0.5)*(rQ/w)^(-1.5)] / (rQ/w)