Differentiate y = x3 from first principles.

To differentiate a function $f(x)$ from first principles, we use the definition of the derivative:

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

i) $y = x^3$

Step 1: Identify $f(x)$ and $f(x+h)$.

$$f(x) = x^3$$ $$f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$$

Step 2: Calculate the difference $f(x+h) - f(x)$.

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