Definition of the Derivative Function and Difference Quotients

1Introduction

This lecture defines the derivative at a point as the limit of average rates of change and constructs the derivative function from these pointwise derivative values.

The line through the fixed point $$(a,f(a\left)\right)$$(a,f(a)) and the point $$(a+h,f(a+h\left)\right)$$(a+h,f(a+h)) on the graph is a secant line, and its slope represents an average rate of change. As $$h\to 0$$h\to0, the latter point approaches the fixed point. If the difference quotient has a finite limit $$f^{\prime }\left(a\right)$$f'(a), the tangent line with that slope is $$y=f\left(a\right)+f^{\prime }\left(a\right)(x-a)$$y=f(a)+f'(a)(x-a).

2Definition

Let $$D\subseteq \mathbb{R}$$D\subseteq\mathbb{R}, $$f:D\to \mathbb{R}$$f:D\to\mathbb{R}, and $$a\in \mathrm{int}D$$a\in\operatorname{int}D. For $$h\ne 0$$h\ne0 satisfying $$a+h\in D$$a+h\in D, the difference quotient at $$a$$a with increment $$h$$h is

Because the expression divides by $$h$$h, $$h=0$$h=0 is excluded. The limit is then taken as $$h\to 0$$h\to0:

If this limit exists as a finite value, then $$f$$f is differentiable at $$a$$a, and $$f^{\prime }\left(a\right)$$f'(a) is the derivative of $$f$$f at $$a$$a. The domain of the derivative function is

The function $$f^{\prime }:D_{f^{\prime }}\to \mathbb{R}$$f':D_{f'}\to\mathbb{R} is the derivative function of $$f$$f.

3Example

As an example, derive the derivative function of $$f:\mathbb{R}\to \mathbb{R}$$f:\mathbb{R}\to\mathbb{R}, $$f\left(x\right)=x^2$$f(x)=x^2, from the definition. For $$h\ne 0$$h\ne0,

Therefore, $$f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }(2x+h)=2x$$f'(x)=\lim_{h\to0}(2x+h)=2x.

This example demonstrates that one must first simplify the expression under the condition $$h\ne 0$$h\ne0 and only afterward take the limit as $$h\to 0$$h\to0.

4Criterion for Differentiability

At an interior point of the domain, the derivative exists if and only if the left and right limits of the difference quotient both exist as finite values and agree. For example, $$\left|x\right|$$|x| is continuous at $$0$$0 but is not differentiable there because of its corner.

5Exercises

6Related Material