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The product rule is a vital concept in calculus that handles the differentiation of functions that are products of two or more expressions. Imagine you have two functions: \( u(x) \) and \( v(x) \), and you want to find the derivative of their product. The product rule helps you achieve this by providing a clear formula:

• \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\).

In this specific problem, \( u(x) = \sqrt{x} \), and \( v(x) = \tan^3(\sqrt{x}) \). To find the derivative, you need to differentiate each part separately and then combine them using the product rule. This organized approach makes it easier to handle complex expressions.
First, differentiate \(u(x)\), and then calculate the derivative of \(v(x)\), which we will discuss further in the following sections. It's crucial to remember you must use the product rule each time you encounter a product of functions, making it a fundamental skill for calculus students.

In calculus, the chain rule is indispensable for finding derivatives of composite functions, which are functions within other functions. Sounds tricky, right? But it's quite manageable. When you differentiate a composite function, the chain rule saves the day. You do this by differentiating the outer function first and then multiplying it by the derivative of the inner function.

• If \( f(x) = g(h(x)) \), then \( \frac{df}{dx} = g'(h(x)) \cdot h'(x) \).

In our exercise, the function \( \tan^3(\sqrt{x}) \) is a perfect example of a composition of functions: "\(\tan^3\)" acts as the outer function, and the inner function is "\(\sqrt{x}\)".
Here you first differentiate \(\tan^3(z)\) with respect to \(z\) as it is the outer part, resulting in \(3\tan^2(z) \cdot \sec^2(z)\). Then multiply by the derivative of \(z = \sqrt{x}\), which is \(\frac{1}{2\sqrt{x}}\).
Combining results gives the derivative of the whole function, central to solving problems involving nested expressions with ease.

The power rule is one of the simplest and most frequently used rules in differentiation. It's incredibly empowering for students needing to find the derivative of any power of \(x\). This rule states that to differentiate a function of the form \(x^n\), you multiply by the exponent and reduce the exponent by one:

• \(\frac{d}{dx}x^{n} = nx^{n-1}\).

In the exercise you're working on, \( u(x) = \sqrt{x} = x^{1/2} \). This means it fits perfectly with the power rule. Applying the rule here, you find \( u'(x) = \frac{1}{2}x^{-1/2} \), which simplifies to \( \frac{1}{2\sqrt{x}} \).
The power rule is not just limited to positive integers. It extends effortlessly to fractions and negative numbers, making it a flexible and robust tool in your differentiation toolkit. Remembering and applying the power rule can significantly reduce the complexity of solving calculus problems.