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Differentiation is a fundamental concept in calculus that deals with finding how a function changes as its input changes. This idea of change is crucial for understanding the behavior of functions. Differentiation helps us:

• Analyze the rate of change of quantities, such as velocity in physics or trends in economics.
• Optimize problems to find maximum or minimum values, useful in fields like engineering and economics.

The process of differentiation involves finding a derivative. It allows us to understand and interpret the slope of the function at any given point. When you differentiate a function, you obtain another function that provides all the slopes (rate of changes) for the original function. Differentiation is also used in many real-world applications, such as predicting population changes or calculating the decline or growth of bank investments.

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input variable. It's like asking, "How does the function respond if we make a tiny tweak to the input?" For the function \( f(x) = \sqrt[3]{x} = x^{1/3} \), its derivative \( f'(x) \) was found using rules of differentiation. In our exercise, the derivative \( f'(x) \) is calculated as:\[f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}}\]The derivative tells you that for a very small change in \( x \), the change in \( f(x) \) is proportional to the derivative. At \( x = 64 \), the derivative \( f'(64) \) is computed and it informs us precisely how the function \( f(x) \) behaves around that point.

The Power Rule is a handy tool in calculus for differentiating expressions where the variable is raised to a power. It simplifies differentiation significantly and is expressed as:If \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).This rule was applied in our exercise for the function \( f(x) = x^{1/3} \). Using the Power Rule, the derivative of \( f(x) \) becomes:\[f'(x) = \frac{1}{3}x^{-2/3}\]This simply means that you bring down the power as a coefficient (which becomes \( \frac{1}{3} \)), and subtract one from the original power \( (1/3) - 1 = -2/3 \). This rule is particularly helpful because it saves time and avoids errors when working with polynomials or any simple power functions.