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In the study of differential equations, understanding the concept of the first derivative is crucial. The first derivative of a function provides the rate at which the function's value changes with respect to a change in its input. Mathematically, for a function like \( f(x) \), the first derivative is denoted as \( f'(x) \).

This first derivative helps us understand the behavior of the function, such as increasing or decreasing trends, and can be calculated using standard differentiation techniques. In situations where functions are often composite, additional rules such as the chain rule become fundamental. First derivatives are used directly in formulating and verifying solutions to differential equations, as seen in the given problem.

Understanding how to correctly compute the first derivative allows us to explore the solution of differential equations deeper and see if the function indeed fits the required equation.

The chain rule is a powerful tool in calculus, particularly when dealing with composite functions, which are functions made up of other functions. It provides a method to calculate the derivative of these composite structures. For a function \( f(g(x)) \), where \( g(x) \) is a function nested within \( f(x) \), the derivative is calculated as \( f'(g(x)) \cdot g'(x) \).

In the problem above, while finding the derivative of \( y_{1}(x) = e^{x/2} \), the chain rule was crucial. Here, \( x/2 \) was our inner function, and \( e^u \) was the outer function, simplifying the derivative to \(\frac{1}{2}e^{x/2} \).

The chain rule is essential not only in calculating these derivatives but also in ensuring the correctness of these computations. It is an indispensable part of solving and verifying differential equations when dealing with nested or composite functions, ensuring every layer of the function is accounted for.

Solution verification is the process of confirming that a proposed solution satisfies the given differential equation. It involves substituting the solution into the equation and checking if both sides of the equation turn out equal. For the differential equation \( 2y' - y = 0 \), this involved substituting the derivatives and original functions to verify equality.

For example, with the function \( y_{1}(x) = e^{x/2} \) and its first derivative \( y_{1}'(x) = \frac{1}{2}e^{x/2} \), substituting these into the differential equation results in both sides simplifying to zero, confirming the solution. But \( y_{2}(x) = x^{2} + 2e^{x/2} \) does not satisfy the equation, showing the substitution does not hold across the same verification process.

Solution verification is an essential part of problem-solving in mathematics, especially in differential equations, as it determines the correctness of the proposed solutions and ensures they adequately represent the model described by the equation.