https://file.aiquickdraw.com/mathbot/file/17297851025747wvffb1k.jpg

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.

If x = sec^2 30 and y = 1 - tan 30, then find dy/dx when y = 4.

Step-by-step solution for finding dy/dx when x = sec^2 30 and y = 1 - tan 30, using trigonometric identities and differentiation techniques. Suitable for high school students in Grades 11-12.

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