The problem in the image is:

Problem (10): $y' = \frac{y}{x}, \text{ then find } \frac{d}{dx}(x^2) = 0$

I will interpret this as a differential equation problem and break it down.

1. The differential equation provided is $y' = \frac{y}{x}$, which can also be written as: $\frac{dy}{dx} = \frac{y}{x}.$

   This is a separable differential equation, which can be solved by separating the variables and integrating both sides.

2. We rewrite it as: $\frac{dy}{y} = \frac{dx}{x}.$

3. Now, integrate both sides: $\int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx,$ which gives: $\ln |y| = \ln |x| + C,$ where $C$ is the constant of integration.

4. Exponentiating both sides to eliminate the logarithms: $|y| = e^{\ln |x| + C} = e^{\ln |x|} e^C = |x| e^C.$

   Let $e^C = C_1$, so we have: $y = C_1 x.$

Thus, the general solution to the differential equation $y' = \frac{y}{x}$ is: $y = C_1 x,$ where $C_1$ is a constant.

Would you like more details or clarifications on any part of this process?

Here are five related questions to consider:

1. How can we apply an initial condition to find the specific solution to this differential equation?
2. What is the significance of separable differential equations in solving real-world problems?
3. How do we interpret the constant of integration $C_1$ in the context of the solution?
4. What are other types of first-order differential equations, and how do they differ from separable ones?
5. How would the solution change if the differential equation were non-linear?

Tip: In solving separable differential equations, always check if variables can be separated and integrated before proceeding with other techniques like substitution.