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The product rule is a fundamental tool in calculus used for differentiating products of two functions. When you have a function that is a product of two other functions like \(f(x) = u(x)v(x)\), the product rule can help find its derivative. The rule is as follows:

• If \(u\) and \(v\) are both functions of \(x\), then the derivative of their product \(uv\) is \(u'v + uv'\).

To apply the product rule, you first need to identify both functions that make up the product and differentiate each one separately. For example, in our exercise, we have \(u = x^2\) and \(v = \tan x\). The product rule tells us that \(f'(x)\) can be found by multiplying \(u'\) (the derivative of \(u\)) by \(v\), and \(u\) by \(v'\) (the derivative of \(v\)), then adding these results together. This step-by-step process ensures you apply the rule correctly to find the derivative of the product.

The power rule is one of the simplest rules of differentiation, perfect for beginners to understand. It's primarily used to differentiate functions that have variables raised to a power. The power rule states:

• If the function is of the form \(x^n\), where \(n\) is any real number, its derivative is \(nx^{n-1}\).

In the given trigonometric function \(x^2 \tan x\), the component \(x^2\) is well-suited for the power rule. By applying the power rule here, we recognize that \(u = x^2\) has a derivative \(u' = 2x\) (because \(2\times x^{2-1} = 2x\)). Using the rule makes it straightforward to find derivatives for functions expressed as power terms, breaking down complex functions into manageable parts.

Differentiation is a cornerstone of calculus, allowing us to determine the rate of change or the slope of a function at any given point. It turns complex functions into simpler derivative forms, which are easier to analyze. In this context, differentiation involves the application of rules like the product rule and power rule to break down and evaluate the derivative of a function.

• Begin by identifying each part of the function you're differentiating.
• Use relevant rules (such as power rule or product rule) to find the derivative of each part.
• Combine these derivatives as needed to form the full derivative.

When differentiating functions like \(f(x) = x^2 \tan x\), you will break it into parts (\(x^2\) and \(\tan x\)), and apply their respective derivative rules (power rule for \(x^2\) and standard differentiation for \(\tan x\)). Then, apply the product rule to bring it all together as explained earlier. Differentiation simplifies the process of analyzing how a function behaves, which is critical in various fields such as physics, engineering, and economics.