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A derivative measures how a function changes as its input changes. For a given function, its derivative is another function that gives the rate of change of the original function with respect to its variable. For instance, if we have a function representing position over time, the derivative of that function would give us the velocity.In our example, we start with the function given by \( y = e^{-0.5 x^2} \). To find the derivative, we need to determine how \( y \) changes with respect to \( x \). This derivative will help us identify the differential equation that the function satisfies. We found that \( y' = -x e^{-0.5 x^2} \). This means that the rate of change of \( y \) with respect to \( x \) is proportional to \( -x \) times the original function itself.

The Chain Rule is a fundamental technique in calculus. It allows us to differentiate composite functions, where one function is inside another. This rule essentially provides a way of computing the derivative of such nested functions.We use the chain rule when we differentiate \( y = e^{-0.5 x^2} \). The inside function is \(-0.5x^2\), and the outside function is the exponential \( e^u \). The chain rule says that to differentiate \( e^{-0.5 x^2} \), we take the derivative of the outside function \( e^u \), which is itself, and multiply it by the derivative of the inside function \(-0.5x^2\). This yields the derivative \( y' = -x e^{-0.5 x^2} \), showing how both components contribute to the rate of change.

Verifying the solution of a differential equation involves checking if the derivative we compute matches with one of the given equations. We need to ensure the function satisfies the equation when substituted.For our function \( y = e^{-0.5 x^2} \), we calculated \( y' = -x e^{-0.5 x^2} \). To verify as a solution, we substitute this into the list of given differential equations and compare the results. When substituted into II, \( y' = -xy \), the left side precisely equals the right side: \(-x (e^{-0.5 x^2}) = -xe^{-0.5 x^2}\). Thus, we've verified that this differential equation is the correct one for the function given the calculated derivative.