gen-define-the-derivative-and-give-one-short-worked-example-5941

Define the derivative and give one short worked example.

# Definition of the Derivative The **derivative** of a function f(x) at a point x is defined as: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This measures the instantaneous rate of change of f at x (or the slope of the tangent line at that point). ## Worked Example Find the derivative of f(x) = x² using the definition. **Solution:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ $$= \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$$ $$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$$ $$= \lim_{h \to 0} \frac{2xh + h^2}{h}$$ $$= \lim_{h \to 0} \frac{h(2x + h)}{h}$$ $$= \lim_{h \to 0} (2x + h)$$ $$= 2x$$ Therefore, **f'(x) = 2x**