I want to ask that as we calculate the derivative at a point by fnding the linear approximation of the region around the point ,does that mean the derivative of points inside the region which we approximate linearly is the same as the derivative the point a
Here we find the derivative at point a by approximating the region around point a by a linear function ,if we try to find the derivative of function at points that are inside the region (a,a+dx) then does the derivative of points inside these region will be the same as derivative of function at point a i.e Is this true: $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}= f'(x_i)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ where $$\ \ x_i\in\mathbb{(x,x+h)}$$