Secant to Tangent: Derivative via First Principles
Consider a function \(f(x)\) and its secant line between \(x\) and \(x+h\). \n\n(a) Calculate the slope of the secant line using the expression \(\frac{f(x+h)-f(x)}{h}\). \n\n(b) Show that as \(h\) approaches 0, the slope of the secant line approaches the derivative \(f'(x)\) using the limit \(\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\). \n\n(c) For \(f(x)=x^2\), use a numerical example […]