Estimate Derivatives Given a Graph - Expii

Given a function graph y=f(x), you can estimate the derivative f'(a) graphically as the slope of the secant line through (a,f(a)) and (a+h,f(a+h)). In principle, as h approaches 0, the estimate will converge to f'(a), as your secant line approaches the tangent line at (a,f(a)). In practice, measurement errors may arise when h is too small. So, which secant line should we choose? How do we balance measurement error with the error from replacing the tangent line with the secant line?.