2.1: Derivatives

We are now ready to explain what is meant by the slope of a curve or the velocity of a moving point. Consider a real function $f$ and a real number $a$ in the domain of $f$. When $x$ has value $a$, $f(x)$ has value $f(a)$. Now suppose the value of $x$ is changed from $a$ to a hyperreal number $a + \Delta x$ which is infinitely close to but not equal to $a$. Then the new value of $f(x)$ will be $f(a + \Delta x)$. In this process the value of $x$ will be changed by a nonzero infinitesimal amount $x$,  while the value of $f(x)$ will be changed by the amount $$f(a + \Delta x) - f(a). \nonumber$$

The ratio of the change in the value of $f(x)$ to the change in the value of $x$ is $$\frac{f(a + \Delta x) - f(a)}{\Delta x}. \nonumber$$

This ratio is used in the definition of the slope of $f$ which we now give.

Definition:

$S$ is said to be the slope of $f$ at $a$ if $$S = st \left(\frac{f(a + \Delta x) - f(a)}{\Delta x}\right) \nonumber$$ for every nonzero infinitesimal $\Delta x$.

The slope, when it exists, is infinitely close to the ratio of the change in $f(x)$ to an infinitely small change in $x$. Given a curve $y = f(x)$, the slope of $f$ at $a$ is also
called the slope of the curve $y = f(x)$ at $x = a$. Figure $\PageIndex{1}$ shows a nonzero infinitesimal $\Delta x$ and a hyperreal straight line through the two points on the curve at $a$ and $a + \Delta x$. The quantity  $$\frac{f(a + \Delta x) - f(a)}{\Delta x} \nonumber$$  is the slope of this line, and its standard part is the slope of the curve.

Figure $\PageIndex{1}$: Hyperreal straight line through a curve.

The slope of $f$ at $a$ does not always exist. Here is a list of all the possibilities.

1. The slope of $f$ at $a$ exists if the ratio  $$\frac{f(a + \Delta x) - f(a)}{\Delta x} \nonumber$$  is finite and has the same standard part for all infinitesimal $\Delta x \neq 0$. It has the value  $$S = st \left(\frac{f(a + \Delta x) - f(a)}{\Delta x}\right). \nonumber$$
2. The slope of $f$ at $a$ can fail to exist in any of four ways:
   1. $f(a)$ is undefined.
   2. $f(a + \Delta x)$ is undefined for some infinitesimal $\Delta x \neq 0$.
   3. The term $\dfrac{f(a + \Delta x) - f(a)}{\Delta x}$ is infinite for some infinitesimal $\Delta x \neq 0$.
   4. The term $\dfrac{f(a + \Delta x) - f(a)}{\Delta x}$ has different standard parts for different infinitesimals $\Delta x \neq 0$.

We can consider the slope of $f$ at any point $x$, which gives us a new function of $x$.

Definition:

Let $f$ be a real function of one variable. The derivative of $f$ is the new function $f'$ whose value at $x$ is the slope of $f$ at $x$. In symbols, $$f'(x) = st \left(\frac{f(x + \Delta x) - f(x)}{\Delta x}\right) \nonumber$$ whenever the slope exists.

The derivative $f'(x)$ is undefined if the slope of $f$ does not exist at $x$.

For a given point $a$, the slope of $f$ at $a$ and the derivative of $f$ at $a$ are the same thing. We usually use the word "slope" to emphasize the geometric picture and
"derivative" to emphasize the fact that $f'$ is a function.

The process of finding the derivative of $f$ is called differentiation. We say that $f$ is differentiable at $a$ if $f'(a)$ is defined; i.e., the slope of $f$ at $a$ exists.

Independent and dependent variables are useful in the study of derivatives. Let us briefly review what they are. A system of formulas is a finite set of equations and inequalities. If we are given a system of formulas which has the same graph as a simple equation $y = f(x)$, we say that $y$ is a function of $x$, or that $y$ depends on $x$, and we call $x$ the independent variable and $y$ the dependent variable.

When $y = f(x)$, we introduce a new independent variable $\Delta x$ and a new dependent variable $\Delta y$, with the equation $$\Delta y = f(x + \Delta x) - f(x).$$

This equation determines $\Delta y$ as a real function of the two variables $x$ and $\Delta x$, when $x$ and $\Delta x$ vary over the real numbers. We shall usually want to use the Equation $(\PageIndex{1})$ for $\Delta y$ when $x$ is a real number and $\Delta x$ is a nonzero infinitesimal. The Transfer Principle implies that Equation $(\PageIndex{1})$ also determines $\Delta y$ as a hyperreal function of two variables when $x$ and $\Delta x$ are allowed to vary over the hyperreal numbers.

$\Delta y$ is called the increment of $y$. Geometrically, the increment $\Delta y$ is the change in $y$ along the curve corresponding to the change $\Delta x$ in $x$. The symbol $y'$ is sometimes used for the derivative, $y' = f'(x)$. Thus the hyperreal equation $$f'(x) = st \left(\frac{f(x + \Delta x) - f(x)}{\Delta x}\right) \nonumber$$ now takes the short form $$y' = st \left(\frac{\Delta y}{\Delta x}\right) \nonumber$$

The infinitesimal $\Delta x$ may be either positive or negative, but not zero. The various possibilities are illustrated in Figure $\PageIndex{2}$ using an infinitesimal microscope. The signs of $\Delta x$ and $\Delta y$ are indicated in the captions.

Figure $\PageIndex{2}$: The possibilities of $\Delta x$ and $\Delta y$ combinations.

Our rules for standard parts can be used in many cases to find the derivative of a function. There are two parts to the problem of finding the derivative $f'$ of a function $f$:

1. Find the domain of $f'$.
2. Find the value of $f'(x)$ when it is defined.

Example $\PageIndex{1}$

Find the derivative of the function $$f(x) = x^{3}. \nonumber$$

Solution

In this and the following examples we let $x$ vary over the real numbers and $\Delta x$ vary over the nonzero infinitesimals. Let us introduce the new variable $y$ with the equation $y = x^{3}$. We first find $\Delta y/\Delta x$.

$$\begin{align*} y &= x^{3}, \\ y + \Delta y &= (x + \Delta x)^{3}, \\ \Delta y &= (x + \Delta x)^{3} - x^{3}, \\ \frac{\Delta y}{\Delta x} &= \frac{(x + \Delta x)^{3} - x^{3}}{\Delta x}. \end{align*}$$

Next we simplify the expression for $\Delta y/\Delta x$.

$$\begin{align*} \frac{\Delta y}{\Delta x} &= \frac{\left(x^{3} + 3x^{2} \Delta x + 3x (\Delta x)^{2} + (\Delta x)^{3}\right) - x^{3}}{\Delta x} \\ &= \frac{3x^{2} \Delta x + 3x(\Delta x)^{2} + (\Delta x)^{3}}{\Delta x} \\ &= 3x^{2} + 3x \Delta x + (\Delta x)^{2}. \end{align*}$$

Then we take the standard part, $$\begin{align*} st \left(\frac{\Delta y}{\Delta x}\right) &= st \left(3x^{2} + 3x \Delta x + (\Delta x)^{2}\right) \\ &= st(3x^{2}) + st(3x \Delta x) + st((\Delta x)^{2}) \\ &= 3x^{2} + 0 + 0. \end{align*}$$

Therefore, $$f'(x) = st \left(\frac{\Delta y}{\Delta x}\right) = 3x^{2}. \nonumber$$

We have shown that the derivative of the function $$f(x) = x^{3} \nonumber$$ is the function $$f'(x) = 3x^{2} \nonumber$$  with the whole real line as domain. $f(x)$ and $f'(x)$ are shown in Figure $\PageIndex{3}$.

Figure $\PageIndex{3}$: Graphs of $y = x^{3}$ and $y' = 3x^{2}$.

Example $\PageIndex{2}$

Find $f'(x)$ given $f(x) = \sqrt{x}$.

Solution

Case 1: $x < 0$

Since $\sqrt{x}$ is not defined, $f'(x)$ does not exist.

Case 2: $x = 0$

When $\Delta x$ is a negative infinitesimal, the term $$\frac{\sqrt{x + \Delta x} - \sqrt{x}}{\Delta x} = \frac{\sqrt{0 + \Delta x} - \sqrt{0}}{\Delta x} \nonumber$$ is not defined because $\sqrt{\Delta x}$ is undefined. When $\Delta x$ is a positive infinitesimal, the term  $$\frac{\sqrt{x + \Delta x} - \sqrt{x}}{\Delta x} = \frac{\sqrt{\Delta x}}{\Delta x} = \frac{1}{\sqrt{\Delta x}} \nonumber$$ is defined but its value is infinite. Thus for two reasons, $f'(x)$ does not exist.

Case 3: $x > 0$

Let $y = \sqrt{x}$. Then $$\begin{align*} y + \Delta y &= \sqrt{x + \Delta x}, \\ \Delta y &= \sqrt{x + \Delta x} - \sqrt{x}, \\ \frac{\Delta y}{\Delta x} &= \frac{\sqrt{x + \Delta x} - \sqrt{x}}{\Delta x}. \end{align*}$$

We then make the computation $$\begin{align*} \frac{\Delta y}{\Delta x} &= \frac{\left(\sqrt{x + \Delta x} - \sqrt{x}\right)}{\Delta x} \cdot \frac{\left(\sqrt{x + \Delta x} + \sqrt{x}\right)}{\left(\sqrt{x + \Delta x} + \sqrt{x}\right)} \\ &= \frac{(x + \Delta x) - x}{\Delta x \left(\sqrt{x + \Delta x} + \Delta x\right)} \\ &= \frac{\Delta x}{\Delta x \left(\sqrt{x + \Delta x} + \sqrt{x}\right)} = \frac{1}{\sqrt{x + \Delta x} + \sqrt{x}}. \end{align*}$$

Taking standard parts, $$\begin{align*} st \left(\frac{\Delta y}{\Delta x}\right) &= st \left(\frac{1}{\sqrt{x + \Delta x} + \sqrt{x}}\right) \\ &= \frac{1}{st \left(\sqrt{x + \Delta x} + \sqrt{x}\right)} \\ &= \frac{1}{st \left(\sqrt{x + \Delta x}\right) + st \left(\sqrt{x}\right)} \\ &= \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2 \sqrt{x}} \end{align*}.$$

Therefore, when $x > 0$, $f'(x) = \dfrac{1}{2 \sqrt{x}}$.

So, the derivative of $f(x) = \sqrt{x}$ is the function $f'(x) = \dfrac{1}{2 \sqrt{x}}$, and the set of all $x > 0$ is its domain (see Figure $\PageIndex{4}$.

Figure $\PageIndex{4}$: Graphs of $y = \sqrt{x}$ and $y' = \frac{1}{2 \sqrt{x}}$.

Example $\PageIndex{3}$

Find the derivative of $f(x) = 1/x$. 

Solution

Case 1: $x = 0$

Then $1/x$ is undefined so $f'(x)$ is undefined.

Case 2: $x \neq 0$

$$\begin{align*} y &= 1/x, \\ y + \Delta y &= \frac{1}{x + \Delta x}, \\ \Delta y &= \frac{1}{x + \Delta x} - \frac{1}{x}, \\ \frac{\Delta y}{\Delta x} &= \frac{1/(x + \Delta x) - 1/x}{\Delta x}. \end{align*}$$

Simplifying, $$\begin{align*} \frac{1/(x + \Delta x) - 1/x}{\Delta x} &= \frac{x - (x + \Delta x)}{x(x + \Delta x) \Delta x} = \frac{-\Delta x}{x(x + \Delta x) \Delta x} \\ &= \frac{-1}{x + \Delta x} \end{align*}$$

Taking the standard part, $$\begin{align*} st(\left(\frac{\Delta y}{\Delta x}\right) &= st \left(-\frac{1}{x(x + \Delta x)}\right) = -\frac{1}{st(x (x + \Delta x))} \\ &= -\frac{1}{st(x) st(x + \Delta x)} = -\frac{1}{x \cdot x} = -\frac{1}{x^{2}}. \end{align*}$$

Thus, $$f'(x) = -\frac{1}{x^{2}}. \nonumber$$

The derivative of the function $f(x) = 1/x$ is the function $f'(x) = - 1/x^{2}$ whose domain is the set of all $x \neq 0$. Both functions are graphed in Figure $\PageIndex{5}$.

Figure $\PageIndex{5}$: Graph of $y = \frac{1}{x}$ and $y' = -\frac{1}{x^{2}}$.

Example $\PageIndex{4}$

Find the derivative of $f(x) = |x|$.

Solution

Case 1 $x > 0$.

In this case $|x| = x$, and we have $$\begin{align*} y &= x, \\ y + \Delta y &= x + \Delta x, \\ \Delta y &= \Delta x, \\ \frac{\Delta y}{\Delta x} &= 1, \quad f'(x) = 1. \end{align*}$$

Case 2: $x < 0$.

Now $|x| = -x$, and $$\begin{align*} y &= -x, \\ y + \Delta y &= -(x + \Delta x), \\ \Delta y &= -(x + \Delta x) - (-\Delta x) = -\Delta x, \\ \frac{\Delta y}{\Delta x} = -\frac{\Delta x}{\Delta x} &= -1, \quad f'(x) = -1. \end{align*}$$

Case 3: $x = 0$.

Then $$\begin{align*} y &= 0, \\ y + \Delta y &= |0 + \Delta x| = |\Delta x|, \\ \Delta y &= |\Delta x|, \\ \frac{\Delta y}{\Delta x} = \frac{|\Delta x|}{\Delta x} &= \begin{cases} 1 &\quad \text{if } \Delta x>0, \\ -1 &\quad \text{if } \Delta x<0. \end{cases} \end{align*}$$

The standard part of $\Delta y/\Delta x$ is then $1$ for some values of $\Delta x$ and $-1$ for others. Therefore $f'(x)$ does not exist when $x = 0$.

In summary, $$f'(x) = \begin{cases} 1 &\quad \text{if } x > 0, \\ -1 &\quad \text{if } x < 0, \\ \text{undefined} &\quad \text{if } x = 0 \end{cases} \nonumber$$

Figure $\PageIndex{6}$ shows $f(x)$ and $f'(x)$.

Figure $\PageIndex{6}$: Graph of $y = |x|$ and its derivative.

The derivative has a variety of applications to the physical, life, and social sciences. It may come up in one of the following contexts.

Velocity: If an object moves according to the equation $s = f(t)$ where $t$ is time and $s$ is distance, the derivative $v = f'(t)$ is called the velocity of the object at
time $t$.

Growth rates: A population $y$ (of people, bacteria, molecules, etc.) grows according to the equation $y = f(t)$ where $t$ is time. Then the derivative $y' = f'(t)$ is
the rate of growth of the population $y$ at time $t$.

Marginal values (economics): Suppose the total cost (or profit, etc.) of producing $x$ items is $y = f(x)$ dollars. Then the cost of making one additional item is approximately the derivative $y' = f'(x)$ because $y'$ is the change in $y$ per unit change in $x$. This derivative is called the marginal cost.

Example $\PageIndex{5}$

A ball thrown upward with initial velocity $b$ ft per sec will be at a height $$y = bt - 16t^{2} \nonumber$$ feet after $t$ seconds. Find the velocity at time $t$. Let $t$ be real and $\Delta t \neq 0$, infinitesimal.

Solution

$$\begin{align*} y + \Delta y &= b(t + \Delta t) - 16(t + \Delta t)^{2}, \\[4pt] \Delta y &= \left[b(t + \Delta t) - 16(t + \Delta t)^{2}\right] - \left[bt - 16t^{2}\right], \\[4pt] \frac{\Delta y}{\Delta t} &= \frac{\left[b(t + \Delta t) - 16(t + \Delta t)^{2}\right] - \left[bt - 16t^{2}\right]}{\Delta t} \\ &= \frac{b \Delta t - 32t \Delta t - 16(\Delta t)^{2}}{\Delta t} \\ &= b - 32t - 16 \Delta t. \end{align*}$$

At time $t$ sec, $$v = y' = b - 32 \ \text{ft/sec}. \nonumber$$

Both functions are graphed in Figure $\PageIndex{7}$.

Figure $\PageIndex{7}$: Graph of a ball's height $y = bt - 16t^{2}$ and its vertical velocity $v = b - 32t$.

Example $\PageIndex{6}$

Suppose a bacterial culture grows in such a way that at time $t$ there are $t^{3}$ bacteria. Find the rate of growth at time $t = 1000$ sec.

Solution

$$y = t^{3}, \quad y' = 3t^{2} \text{ by Example } \PageIndex{1}. \nonumber$$

At $t = 1000$ sec, $$y' = 3,000,000 \ \text{bacteria/sec}. \nonumber$$

Example $\PageIndex{7}$

Suppose the cost of making $x$ needles is $\sqrt{x}$ dollars. What is the marginal cost after 10,000 needles have been made?

Solution

$$y = \sqrt{x}, \quad y' = \frac{1}{2 \sqrt{x}} \text{ by Example } \PageIndex{2}. \nonumber$$

At $x = 10,000$, $$y' = \frac{1}{2 \sqrt{10,000}} = \frac{1}{200} \ \text{dollars per needle}. \nonumber$$

Thus the marginal cost is one half of a cent per needle.

Problems for Section 2.1

Find the derivative of the given function in Problems 1-21.

1.	$f(x) = x^{2}$	2.	$f(t) = t^{2} + 3$
3.	$f(x) = 1 - 2x^{2}$	4.	$f(x) = 3x^{2}$
5.	$f(t) = 4t$	6.	$f(x) = 2 - 5x$
7.	$f(t) = 4t^{3}$	8.	$f(t) = -t^{3}$
9.	$f(u) = 5 \sqrt{u}$	10.	$f(u) = \sqrt{u + 2}$
11.	$g(x) = x \sqrt{x}$	12.	$g(x) = 1 / \sqrt{x}$
13.	$g(t) = t^{-2}$	14.	$g(t) = t^{-3}$
15.	$f(y) = 3y^{-1} + 4y$	16.	$f(y) = 2y^{3} + 4y^{2}$
17.	$f(x) = ax + b$	18.	$f(x) = ax^{2}$
19.	$f(x) = \sqrt{ax + b}$	20.	$f(x) = 1/(x + 2)$
21.	$f(x) = 1/(3 - 2x)$
22.	Find the derivative of $f(x) = 2x^{2}$ at the point $x = 3$.
23.	Find the slope of the curve $f(x) = \sqrt{x-1}$ at the point $x = 5$.
24.	An object moves according to the equation $y = 1/(t + 2)$, $t \geq 0$. Find the velocity as a function of $t$.
25.	A particle moves according to the equation $y = t^{4}$. Find the velocity as a function of $t$.
26.	Suppose the population of a town grows according to the equation $y = 100t + t^{2}$. Find the rate of growth at time $t = 100$ years.
27.	Suppose a company makes a total profit of $1000x - x^{2}$ dollars on $x$ items. Find the marginal profit in dollars per item when $x = 200$, $x = 500$, and $x = 1000$.
28.	Find the derivative of the function $f(x) = |x + 1|$.
29.	Find the derivative of the function $f(x) = |x^{3}|$.
30.	Find the slope of the parabola $y = ax^{2} + bx + c$ where $a$, $b$, $c$ are constants.