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The concept of a derivative is central to calculus and helps us understand how a function changes at any point. Essentially, the derivative gives the slope of the tangent line to the function at a given point. It shows us the rate at which the function's value changes. The calculation of a derivative involves taking an original function and determining its rate of change. Typically, if you have a function like \(f(x)\), its derivative is denoted \(f'(x)\) or \(\frac{d}{dx}[f(x)]\).
For the function \(f(x)=x^{2/3}(5-2x)\), finding the derivative involves using calculus rules. You apply these rules to reveal new insights about the function, especially how it behaves over an interval.

Critical points are important in determining where a function might attain its minimum or maximum values. A critical point occurs when the derivative of the function is zero or undefined. It indicates places where the slope of the function flattens out. Finding these points is key to understanding the function's behavior and to locating potential extremum points within a given interval.
To find a critical point, you set the derivative equal to zero and solve for the variable. In our example, this involves solving the equation \((10/3)x^{-1/3} - (2/3)x^{1/3} - 2x^{2/3} = 0\). By solving this equation, you find values of \(x\) where the function might reach a peak or trough, which are the potential maximum or minimum values within the interval.

In calculus, the product rule is a fundamental technique used to find the derivative of a product of two functions. If you have two functions, say \(u(x)\) and \(v(x)\), and you need to differentiate \(u(x) \cdot v(x)\), the product rule says:

• The derivative of \(u(x) \cdot v(x)\) is \(u'(x)v(x) + u(x)v'(x)\).

This rule is critical when working with functions that are multiplied together.
For the function \(f(x)=x^{2/3}(5-2x)\), we have two parts: \(x^{2/3}\) and \(5-2x\). Applying the product rule means you differentiate each part, apply the rule, and then simplify. This makes complex derivatives manageable and allows us to explore functions composed of multiple elements easily.