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The **Product Rule** is an essential tool in calculus for differentiating products of functions. If you have two functions multiplied together, say \( u(x) \) and \( v(x) \), their derivative is given by the formula:

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

This formula states that you differentiate the first function and multiply it by the second function, add it to the first function multiplied by the derivative of the second function. Use it whenever you see a product of different functions—like in our example, where we had \( x^2 [f(x)]^3 \). By applying this rule, you're able to break down complex differentiation problems into manageable steps. Separating and individually handling each component of the product makes the task straightforward and systematic.

When differentiating compositions of functions, the **Chain Rule** is your go-to technique. This rule is particularly handy when dealing with complex functions nested within each other.

• If you have a composite function \( y = g(f(x)) \), the derivative is: \[ \frac{dy}{dx} = g'(f(x)) \cdot f'(x). \]

The Chain Rule helps you find the derivative of outer functions that have an inner function. In the exercise, we saw \( [f(x)]^3 \) as part of the term. To differentiate this, we first take the derivative of the outer function \( ([something]^3) \) and then multiply it by the derivative of \( f(x) \). In practical situations, always remember to identify which function is "inside" and which is "outside," and then apply the Chain Rule to simplify computations effectively.

Understanding **function derivatives** is foundational in calculus. A derivative represents an equation's rate of change at any given point. Differentiation allows us to see how a function behaves, changes direction, or achieves extremum points.

• For a basic function \( f(x) \), the derivative \( f'(x) \) gives us insights into its slope or instantaneous rate of change.
• Standard rules, like the power rule, help simplify the differentiation process:\[ \frac{d}{dx} [x^n] = nx^{n-1}. \]

In our exercise, differentiating \( f(x) \) was straightforward, resulting in the derivative \( f'(x) \). The challenge in finding function derivatives often involves applying different rules together—such as combining the Product and Chain Rules—to tackle more elaborate expressions and achieve the solution for given variable values, like finding \( f'(1) \).

**Calculus problem solving** is an art of systematically applying mathematical tools to solve complex differential equations. This involves recognizing what rules and methods apply to different parts of the equation. In our task:

• First, identify constants, single functions, products, and composite functions within the equation.
• Apply appropriate differentiation rules, such as the product rule for multiplied functions and the chain rule for compositions.

In the equation \( f(x) + x^2 [f(x)]^3 = 10 \), careful consideration of these rules allowed us to solve for the unknown derivative \( f'(1) \). The process iteratively builds upon differentiated terms by substituting known values and simplifying expressions to achieve the solution. By combining rules methodically, you accurately isolate and solve each part of the equation step-by-step. This structured approach ensures solutions to even the most daunting calculus problems are clear and achievable.