§1.4 The derivative function

Introduction

How the derivative is itself a function

Summary

Let $f(x) = 4x - x^2$. The limit definition gives $f'(x) = 4 - 2x$. What is $f'(3)$?

By the limit definition, $f(x) = 1$ has $f'(x) = 0$ at every $x$, and $g(t) = t$ has $g'(t) = 1$ at every $t$.

Using the limit definition, what is the derivative of $p(z) = z^2$?

The limit definition gives $q'(s) = 3s^2$ for $q(s) = s^3$. Compute $q'(2)$.

Using the limit definition, what is the derivative of $F(t) = \frac{1}{t}$?

The limit definition gives $G'(y) = \frac{1}{2\sqrt{y}}$ for $G(y) = \sqrt{y}$. Compute $G'(4)$ as a decimal.

A differentiable function satisfies $f(-2) = 1$ with $f'(-2) = -2$, $f'(-1) = -1$, $f'(0) = 0$, $f'(1) = 1$, $f'(2) = 2$. Which formula is consistent with all of this data?

Let $g(x) = x^2 - x + 3$. What is $g'(x)$?

For $p(x) = 5x^2 - 4x + 12$ the limit definition gives $p'(x) = 10x - 4$. Comparing with $g'(x) = 2x - 1$ for $g(x) = x^2 - x + 3$, which constants affected the derivative formula?

Given the graph of a function $y = f(x)$, where does the graph of its derivative $y = f'(x)$ cross the $x$-axis, and how does its sign behave?

If $g'$ is constant on an interval, then $g$ is linear on that interval.