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.