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Differentiating a function that is the sum of two or more functions can be easier than it looks. The sum rule states that the derivative of the entire function is just the sum of the derivatives of each part. For example, if we have a function \( y = f(x) + g(x) \), then the derivative \( y' = f'(x) + g'(x) \). In the exercise, we applied the sum rule to differentiate \( (x-1)^3 \) and \( (x+2)^4 \) separately. This simplifies the process significantly.

When dealing with composite functions, such as \( (x-1)^3 \) or \( (x+2)^4 \), the chain rule is extremely useful. This rule helps us differentiate functions within functions. Mathematically, the chain rule states that \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x) \). For instance, in our exercise, to differentiate \( (x-1)^3 \), we set \( g(x) = x-1 \) and \( f(u) = u^3 \). Applying the chain rule:
- First, find the derivative of the outer function: \( f'(u) = 3u^2 \).
- Then, differentiate the inner function: \( g'(x) = 1 \).
- Multiply these derivatives: \( 3(x-1)^2 \times 1 = 3(x-1)^2 \).
This method simplifies our calculations significantly.

A derivative is a fundamental concept in calculus that represents the rate at which a function is changing at any given point. It can be thought of as the slope of the tangent line to the function's graph at a certain point. In mathematical notation, the derivative of a function \( y = f(x) \) with respect to \( x \) is represented as \( f'(x) \) or \( \frac{dy}{dx} \).
In our exercise, the endpoints of this process were:
- Differentiating \( (x-1)^3 \) produced \( 3(x-1)^2 \).
- Differentiating \( (x+2)^4 \) yielded \( 4(x+2)^3 \).
- Combining these results using the sum rule gave us the final derivative: \( y' = 3(x-1)^2 + 4(x+2)^3 \).
Understanding these rules together simplifies complex differentiation problems.