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In differential calculus, the Power Rule is a basic principle used for finding the derivative of a function of the form \(t^n\). When you encounter an expression like \(f(t) = t^{1/3} - t^{-1/3}\), the Power Rule comes in handy to derive it easily. The Power Rule states that the derivative of \(t^n\), where \(n\) is any real number, is \(n \cdot t^{n-1}\). To differentiate \(t^{1/3}\), follow these steps:

• Identify \(n\) as \(1/3\) from \(t^{1/3}\).
• Apply the Power Rule: \(\frac{d}{dt}[t^{1/3}] = \frac{1}{3} \cdot t^{1/3-1}\).
• This simplifies to \(\frac{1}{3}t^{-2/3}\).

Similarly, for \(t^{-1/3}\):

• Identify \(n\) as \(-1/3\).
• Apply the Power Rule: \(\frac{d}{dt}[t^{-1/3}] = -\frac{1}{3} \cdot t^{-1/3-1}\).
• This simplifies to \(-\frac{1}{3}t^{-4/3}\).

The Power Rule works effectively for differentiating polynomial expressions quickly, making it one of the most frequently used techniques in calculus.

A derivative represents how a function changes as its input changes. It shows the rate of change or the slope of a curve at any point along its length. In the context of the given problem, finding the derivative \(\frac{d}{dt}f(t)\) means calculating how the function \(f(t) = \sqrt[3]{t} - \frac{1}{\sqrt[3]{t}}\) changes concerning \(t\).By expressing each term in the function using exponents, the function becomes \(f(t) = t^{1/3} - t^{-1/3}\). Using the Power Rule, we calculate the derivative as follows:

• The derivative of the first term \(t^{1/3}\) is \(\frac{1}{3}t^{-2/3}\).
• The derivative of the second term \(t^{-1/3}\) is \(-\frac{1}{3}t^{-4/3}\).

Combining these results gives the entire derivative of the function:\[f'(t) = \frac{1}{3}t^{-2/3} + \frac{1}{3}t^{-4/3}\]This expression describes precisely how \(f(t)\) behaves in terms of \(t\), providing crucial insights for analyzing the function's behavior, such as identifying increasing or decreasing intervals.

Differential Calculus is a field in mathematics focused on how functions change, and the derivatives that represent these changes. It provides tools to measure how a quantity occurs over time or under various conditions. By studying derivatives, differential calculus allows us to understand and predict complex systems.The task of differentiation involves finding the derivative, a central theme in differential calculus. For instance, in our exercise where \(f(t)\) is defined, applying differential calculus principles helps in finding \(f'(t)\). This concept of differentiation helps reveal deeper insights:

• Predictive Analysis: Derivatives help predict future values based on current data trends.
• Optimization: Identifying maximum or minimum values of functions is possible using derivatives.

In practical applications, differential calculus underlies fields like physics, economics, and engineering, where change dynamics play a vital role. Calculating the derivative effectively, as shown in our original problem, stems from understanding the relationship between a function's rate of change and its overall behavior, governed by the precise rules of differentiation.