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Calculus is a fascinating field of mathematics focused on change. It explores how things grow and shrink, and how they move along curves. A critical part of calculus is differentiation.

Differentiation is all about finding the rate at which something changes. If you've ever wondered how fast a car is speeding up, you've dabbled with calculus concepts. At its core, calculus deals with two main operations: differentiation and integration. Differentiation gives us the rate of change, while integration accumulates values.

In our example, where the function is given as \(y = 2x^3\), we want to explore how the function behaves as \(x\) changes. Calculus provides the tools to give a precise description of this change. Understanding this foundation will help make the more complex ideas in calculus seem less daunting.

The derivative is a fundamental concept in calculus that describes how a function changes as its input changes. It essentially tells us the "speed" at which our function is changing at any given point.

Formally, the derivative of a function \(f\) at a point \(x\) is defined as the limit of the average rate of change of the function as the interval gets infinitesimally small. The mathematical formula is:

• \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

This formula might look intimidating, but it's just about finding the slope of the curve at a single point.

In practice, calculating a derivative involves substituting the function into this formula, expanding it, simplifying, and then applying the limit process. With \(y = 2x^3\), we plug it into the definition to see how \(y\) changes as \(x\) grows or shrinks slightly. The process of working through this calculation is how we find the derivative.

Finding a derivative might seem like a maze at first, but breaking it into clear steps can help. Here's how you go about differentiating a function like \(y = 2x^3\) using its definition.

1. **Start with the formula!**
The derivative definition is your template. Use \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\) as your guide.

2. **Substitute the function.**
Replace \(f(x)\) with your function, \(2x^3\). This means you're setting up the problem: \( \lim_{h \to 0} \frac{2(x+h)^3 - 2x^3}{h} \).

3. **Expand and simplify.**
Use algebra to expand \((x+h)^3\), then simplify. Here, \((x+h)^3\) becomes \(x^3 + 3x^2h + 3xh^2 + h^3\), leading to an expression you can simplify.

4. **Factor and cancel.**
This step is crucial. You'll factor out \(h\) from the numerator to allow cancelling of terms. The numerator simplifies to \(h(6x^2 + 6xh + 2h^2)\).

5. **Evaluate the limit.**
Finally, as \(h\) approaches zero, all the terms containing \(h\) disappear, and the expression boils down to \(6x^2\). This is the derivative.

• As a result, the derivative of \(y = 2x^3\) is \(y' = 6x^2\).

Understanding these steps helps demystify the process and makes taking derivatives with this method more straightforward.