1. The problem is to find the derivative of a function using the first principle of calculus, also known as the definition of the derivative. 2. The formula for the derivative of a function $f(x)$ at a point $x$ using the first principle is: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This means we calculate the difference quotient and then take the limit as $h$ approaches zero. 3. Important rules: - The limit must exist for the derivative to exist. - Simplify the expression inside the limit before taking the limit. 4. Example: Find the derivative of $f(x) = x^2$ using the first principle. 5. Calculate $f(x+h)$: $$f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$$ 6. Substitute into the difference quotient: $$\frac{f(x+h) - f(x)}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h}$$ 7. Simplify by canceling $h$: $$\frac{\cancel{h}(2x + h)}{\cancel{h}} = 2x + h$$ 8. Take the limit as $h \to 0$: $$\lim_{h \to 0} (2x + h) = 2x$$ 9. Therefore, the derivative of $f(x) = x^2$ is: $$f'(x) = 2x$$ This method can be applied to find the derivative of any differentiable function using the first principle.