14.7: Infinitesimal calculus for derivatives

This chapter discussing the calculus of differentiation in a very brief manner. Proofs are limited and applications are more emphasized here. Most of the sections here are from other LibreTexts courses/books rather than the main authors of this course. Tables, applications, and tidbits are herein and as such this does not replace a calculus course where you will learn most of the theory properly. Here we are really only concerned with how derivatives are used in engineering and science. While the basics should be understood in this chapter, it is not necessary to understand the full scope presented here as you will have that opportunity in calculus class proper.

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

We previously defined a derivative in the sense of an application derivative. Not a complete true definition but a practical one.

\[v = \frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}\]

For a true definition of derivative we need to look to the fundamental theorems of calculus, but we can still improve on the practical derivative by making it general.

\[\dfrac{df(x)}{dx} = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}\]

This is akin to the definition you would see in numerical methods, though various "improvements" are done to ensure a better answer since we can't actually have the "step size" go to zero.

For an example we will use the previous derivative definition to solve for the \(\sqrt{x}\).

\[f(x) = \sqrt{x}\]

\[\dfrac{df(x)}{dx} = \lim_{h \to 0} \dfrac{\sqrt{x+h}-\sqrt{x}}{h}\]

\[\dfrac{df(x)}{dx} = \lim_{h \to 0} \dfrac{\sqrt{x+h}-\sqrt{x}}{h}\]

Now let us do a standard trick of math: \((a+b)(a-b) = a^2 - b^2\). Where we square the square roots and "remove" the middle term which would still have square roots if we did not use this trick.

\[\dfrac{df(x)}{dx} = \lim_{h \to 0} \dfrac{\sqrt{x+h}-\sqrt{x}}{h} \dfrac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\]

\[\dfrac{df(x)}{dx} = \lim_{h \to 0} \dfrac{(x+h)-(x)}{h \left(\sqrt{x+h}+\sqrt{x}\right)}\]

\[\dfrac{f(x)}{dx} = \lim_{h \to 0} \dfrac{h}{h \left(\sqrt{x+h}+\sqrt{x}\right)}\]

\[\dfrac{f(x)}{dx} = \lim_{h \to 0} \dfrac{1}{\sqrt{x+h}+\sqrt{x}}\]

Now we can take the limit of h going to zero easily but just setting it to zero. The final answer is

\[\dfrac{d (\sqrt{x})}{dx} = \dfrac{1}{2 \sqrt{x}}\]

Is this right? Let's try this again with the standard table definition of how to do this: that is multiply by the exponent and then subtract one from the exponent.

\[\dfrac{d(x^{\frac{1}{2}})}{dx}\]

\[\dfrac{1}{2} x^{-\frac{1}{2}}\]

Which is the same as the solution using the limit...ok looks good. Let's start looking at some applications, but first let us look at a standard table of derivatives.

• 14.7.1: Table of Derivatives

  Table of derivatives.

• 14.7.2: Derivatives of Exponential and Logarithmic Functions

  In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.

• 14.7.3: Maxima and Minima

  Finding the maximum and minimum values of a function  has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.

• 14.7.4: Derivatives and the Shape of a Graph

  Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.

• 14.7.5: L’Hôpital’s Rule

  In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.

• 14.7.6: Newton’s Method

  In many areas of pure and applied mathematics, we are interested in finding solutions to an equation of the form f(x)=0. For most functions, however, it is difficult—if not impossible—to calculate their zeroes explicitly. In this section, we take a look at a technique that provides a very efficient way of approximating the zeroes of functions. This technique makes use of tangent line approximations and is behind the method used often by calculators and computers to find zeroes.