https://file.aiquickdraw.com/mathbot/file/17306945127127pwxiki4.jpg

The function in the image is:

\[
Q = 10 L^{0.7} K^{0.3}
\]

To find the first-order partial derivatives of \( Q \) with respect to \( L \) and \( K \), we'll treat each variable independently.

### 1. Partial Derivative of \( Q \) with respect to \( L \), denoted \( \frac{\partial Q}{\partial L} \):
Since \( Q = 10 L^{0.7} K^{0.3} \), applying the power rule to \( L^{0.7} \) while treating \( K \) as a constant:

\[
\frac{\partial Q}{\partial L} = 10 \cdot 0.7 L^{0.7 - 1} K^{0.3}
\]

Simplifying this expression:

\[
\frac{\partial Q}{\partial L} = 10 \cdot 0.7 L^{-0.3} K^{0.3}
\]
\[
\frac{\partial Q}{\partial L} = 7 L^{-0.3} K^{0.3}
\]

### 2. Partial Derivative of \( Q \) with respect to \( K \), denoted \( \frac{\partial Q}{\partial K} \):
Similarly, we differentiate \( Q = 10 L^{0.7} K^{0.3} \) with respect to \( K \), treating \( L \) as a constant:

\[
\frac{\partial Q}{\partial K} = 10 \cdot 0.3 L^{0.7} K^{0.3 - 1}
\]

Simplifying this expression:

\[
\frac{\partial Q}{\partial K} = 10 \cdot 0.3 L^{0.7} K^{-0.7}
\]
\[
\frac{\partial Q}{\partial K} = 3 L^{0.7} K^{-0.7}
\]

### Summary of Partial Derivatives
The first-order partial derivatives are:
\[
\frac{\partial Q}{\partial L} = 7 L^{-0.3} K^{0.3}
\]
\[
\frac{\partial Q}{\partial K} = 3 L^{0.7} K^{-0.7}
\]

Would you like a further breakdown of these steps or have any specific questions?

Here are some additional questions you might find useful:
1. What is the interpretation of these partial derivatives in economic terms?
2. How would these partial derivatives change if the exponents of \( L \) and \( K \) were different?
3. How do second-order partial derivatives of \( Q \) with respect to \( L \) and \( K \) look like?
4. What happens to \( \frac{\partial Q}{\partial L} \) if we increase \( K \) while keeping \( L \) constant?
5. How could you use these partial derivatives to determine the marginal product of labor and capital?

**Tip:** When differentiating functions with constants, focus on applying the power rule to variables of interest while treating other variables as constants.

Find the first order partial derivatives of Q = 10L^0.7 K^0.3

This page provides a step-by-step solution for finding the first-order partial derivatives of the function Q = 10L^0.7 K^0.3 with respect to L and K. The solution covers fundamental calculus concepts, including partial derivatives and the power rule, ideal for college-level calculus students.

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