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When differentiating products of two functions, it's essential to use the Product Rule. This rule simplifies finding a derivative when the function is a multiplication of two different functions. For example, if you have two functions, \( u(x) \) and \( v(x) \), the Product Rule states that the derivative of their product \( (uv)' \) is given by: \[ (uv)' = u'v + uv' \] This means that you first differentiate the first function \( u \), multiply it by the second function \( v \), and vice versa. Then, you add these two products together. It transforms a complex differentiation problem into manageable pieces. In our exercise, this rule was instrumental in differentiating the product of \( \sqrt{x} \) and \( x^{4/3} + 3 \).

Understanding how to find the derivative of common functions makes calculus much easier. Derivatives are, in essence, the way to find how a function changes at any given point. Let's break down two essential functions and their derivatives from our exercise:

• For \( u(x) = \sqrt{x} \), its derivative is \( u'(x) = \frac{1}{2\sqrt{x}} \). This derivative indicates how \( \sqrt{x} \) changes with respect to \( x \).
• For \( v(x) = x^{4/3} + 3 \), its derivative is \( v'(x) = \frac{4}{3}x^{1/3} \). This derivate shows how both terms within the function contribute to its rate of change.

Identifying and understanding these derivatives is crucial because they form the basis for applying other rules, like the Product Rule, which was used to solve our exercise.

Mathematical notation is a powerful language that allows mathematicians to succinctly express complex ideas in a structured form. In calculus, clear and accurate notation is critical to solving problems correctly. Here are some notations used:

• \( f(x) \) and \( f'(x) \): \( f(x) \) represents a function of \( x \), and \( f'(x) \) denotes its derivative.
• \( u(x) \), \( v(x) \) and their derivatives \( u'(x) \), \( v'(x) \): These notations help categorize different functions and their derivatives, especially when applying rules like the Product Rule.
• Exponents such as \( x^{4/3} \): Indicate the power to which \( x \) is raised and are integral in describing rates of change within functions.

This notation keeps mathematical expressions neat and intelligible, ensuring efficient communication and application of calculus concepts.