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Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function is changing at any given point. In simpler terms, differentiation tells you how a function's output changes as its input changes. This is crucial for understanding how things grow and change over time. For example, in physics, differentiation helps determine velocity and acceleration. In our exercise, differentiation is used to find the second derivative of the function.

The power rule is one of the most straightforward rules in differentiation. It states that if you have a function of the form \(f(x) = x^n\), then the derivative \(f'(x)\) is given by \(f'(x) = nx^{n-1}\). This rule is highly useful because it simplifies the process of finding the derivative.
Remember, you just bring down the exponent as a coefficient and then reduce the exponent by one. In the given exercise, we apply the power rule to each term separately:

• For \(2x^{-3}\), the derivative is \(2(-3)x^{-4}\)
• For \(x^{-2}\), the derivative is \(-2x^{-3}\).

These simple steps make differentiation very manageable.

Higher-order derivatives are the derivatives of derivatives. Essentially, after you find the first derivative of a function, you can find the second, third, and so on. These higher-order derivatives provide more detailed information about the function's behavior.
For example, while the first derivative represents the rate of change, the second derivative represents the rate at which the rate of change is itself changing.
In the given exercise, we first simplified the function \(y\) to a more easily differentiable form. Then, we found the first derivative \(y'\). By differentiating \(y'\), we obtained the second derivative \(y''\). In our case, the second derivative is given by: \(y'' = 24x^{-5} + 6x^{-4}\). This gives us deeper insight into the behavior and concavity of the original function \(y\).