To calculate $\frac{dP}{dQ}$ for the function

$P = 64Q - (200Q - 767)^{1.56} \cdot 100$

we need to differentiate $P$ with respect to $Q$ and then evaluate it at $Q = 15$.

1. Differentiate $P$:

Using the chain rule and product rule, we find:

$\frac{dP}{dQ} = 64 - 100 \cdot 1.56 \cdot (200Q - 767)^{0.56} \cdot \frac{d}{dQ}(200Q - 767)$

$\frac{d}{dQ}(200Q - 767) = 200$

Thus,

$\frac{dP}{dQ} = 64 - 100 \cdot 1.56 \cdot (200Q - 767)^{0.56} \cdot 200$

2. Evaluate at $Q = 15$:

First, calculate $200Q - 767$ at $Q = 15$:

$200(15) - 767 = 3000 - 767 = 2233$

Now substitute $Q = 15$ into the derivative:

$\frac{dP}{dQ} = 64 - 100 \cdot 1.56 \cdot (2233)^{0.56} \cdot 200$

Calculating $(2233)^{0.56}$:

$(2233)^{0.56} \approx 16.57 \quad (\text{using a calculator})$

Now substitute this value back:

$\frac{dP}{dQ} = 64 - 100 \cdot 1.56 \cdot 16.57 \cdot 200$

Calculating the multiplication:

$100 \cdot 1.56 \cdot 16.57 \cdot 200 \approx 516800$

So,

$\frac{dP}{dQ} = 64 - 516800$

Calculating this:

$\frac{dP}{dQ} \approx -516736$

3. Final Rounded Answer:

Thus, the value of $\frac{dP}{dQ}$ when $Q = 15$ is approximately:

$\frac{dP}{dQ} \approx -516736.00$

Summary

The value of $\frac{dP}{dQ}$ at $Q = 15$ is $-516736.00$.

Would you like to go over any details or have any questions? Here are some related questions for further exploration:

1. What are the implications of the derivative value in terms of the function's behavior?
2. How would the derivative change if the exponent were different?
3. Can you explain the significance of the chain rule in this context?
4. What happens to the derivative if $Q$ approaches a different value?
5. How would you interpret this derivative in a real-world scenario?

Tip: When dealing with derivatives, always ensure to apply the correct differentiation rules to avoid errors in calculation.