§1.4 The derivative function

Introduction

How the derivative is itself a function

Summary

Consider the function $f(x) = 4x - x^2$.

Use the limit definition to compute each of the following derivative values: $f'(0)$, $f'(1)$, $f'(2)$, and $f'(3)$.

$f(x) = 1$

$g(t) = t$

$p(z) = z^2$

$q(s) = s^3$

$F(t) = \dfrac{1}{t}$

$G(y) = \sqrt{y}$

Let $f$ be a function with the following properties: $f$ is differentiable at every value of $x$ (that is, $f$ has a derivative at every point), $f(-2) = 1$, and $f'(-2) = -2$, $f'(-1) = -1$, $f'(0) = 0$, $f'(1) = 1$, and $f'(2) = 2$. - On the axes provided at left in Figure 1.4, sketch a possible graph of $y = f(x)$. Explain why your graph meets the stated criteria. - Conjecture a formula for the function $y = f(x)$. Use the limit definition of the derivative to determine the corresponding formula for $y = f'(x)$. Discuss both graphical and algebraic evidence for whether or not your conjecture is correct.

Consider the function $g(x) = x^2 - x + 3$. - Use the limit definition of the derivative to determine a formula for $g'(x)$. - Use a graphing utility to plot both $y = g(x)$ and your result for $y = g'(x)$; does your formula for $g'(x)$ generate the graph you expected? - Use the limit definition of the derivative to find a formula for $p'(x)$ where $p(x) = 5x^2 - 4x + 12$. - Compare and contrast the formulas for $g'(x)$ and $p'(x)$ you have found. How do the constants 5, 4, 12, and 3 affect the results?

For each graph that provides an original function $y = f(x)$ in Figure 1.4, your task is to sketch an approximate graph of its derivative function, $y = f'(x)$, on the axes immediately below. View the scale of the grid for the graph of $f$ as being $1 \times 1$, and assume the horizontal scale of the grid for the graph of $f'$ is identical to that for $f$. If you need to adjust the vertical scale on the axes for the graph of $f'$, you should label that accordingly.

Let $g$ be a continuous function (that is, one with no jumps or holes in the graph) and suppose that a graph of $y= g'(x)$ is given by the graph on the right in Figure 1.4.

• Observe that for every value of $x$ that satisfies $0 \lt x \lt 2$, the value of $g'(x)$ is constant. What does this tell you about the behavior of the graph of $y = g(x)$ on this interval? - On what intervals other than $0 \lt x \lt 2$ do you expect $y = g(x)$ to be a linear function? Why? - At which values of $x$ is $g'(x)$ not defined? What behavior does this lead you to expect to see in the graph of $y=g(x)$? - Suppose that $g(0) = 1$. On the axes provided at left in Figure 1.4, sketch an accurate graph of $y = g(x)$.