Derivatives of Algebraic Functions

DERIVATIVES OF ALGEBRAIC FUNCTIONS

An algebraic function is any function that can be built from the identity function y=x by forming linear combinations, products, quotients, and fractional powers.

Examples. All of the following are algebraic functions.

f(x) = sqrt(x) = x^1/2
g(x) = |x| = sqrt(x²)
h(x) = sqrt(|x|) = sqrt(sqrt(x²))

Algebraic functions can have rather interesting graphs, such as the following "seagull graph."

**y = sqrt(|x|)**

Using the linearity of the derivative, the product rule, the quotient rule and the power rule, we can differentiate any algebraic function. The derivative of an algebraic functions is another algebraic function.

Example. Differentiate the absolute value function.

Solution. A glance at the graph of the absolute value function should convince you that the numbers 1 and -1 should be values of the derivative.

**y = |x|**

This suspicion is easily verified. Remember that |x| = sqrt(x²) for all real numbers x. We compute the derivative as follows.

|x|

sqrt(x²)

(

2sqrt(x²)

)

(2x)

|x|

{	1	if x>0
{	-1	if x<0

Did you see the application of the chain rule in the first line of this calculation? As we discussed on the page "Differentiability vs. Continuity," the absolute value function is not differentiable at 0. The derivative is not defined at zero.

There are two special classes of algebraic functions that should be mentioned.

You can consult the Field Guide to Functions for the basic concepts, notation and terminology of polynomial functions.

Polynomial Functions.

A polynomial function is a function that arises as a linear combination of a constant function and any finite number of power functions with positive integer exponents. A generic polynomial has the following form.

p(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀
The largest integer power n that appears in this expression is the degree of the polynomial function. A polynomial function of degree one (respectively two, three, four, five) is a linear (respectively quadratic, cubic, quartic, quintic) polynomial function.

Combining the power rule and the linearity of the derivative, one notes that the derivative of a polynomial of degree n is a polynomial of degree n-1. The second derivative of a polynomial of degree n is a polynomial of degree n-2. And so on. The (n+1)st derivative of a polynomial of degree n is the zero function: p⁽ⁿ⁺¹⁾(x) = 0. In Stage 9, we will see that this property distinguishes polynomial functions as a class.

Any function whose (n+1)st derivative is the zero function is a polynomial of degree at most n.

For example, any function with a constant (or degree zero) derivative is linear (y=mx+b). Any function with a linear derivative is quadratic, and so on. None of these statements is at all obvious at this point. In order to verify this we will need th Mean Value Theorem. Stay tuned.

To illustrate the significance of statements such as this, we note that these facts have a lot to do with the way in which projectiles fly through the air. For in the absence of significant frictional forces, the acceleration that acts on a projectile near the surface of the earth is constant (at approximately 32 ft/sec²). Since acceleration is the second derivative of position (the first derivative of position is velocity), this means that the position of a projectile (for example, its height) is described by a quadratic function. This provides us with a reliable means for making quantitative and precise assessments and predictions that involve a basic physical phenomenon (the flight of a projectile). This is the kind of success that science is really all about.

It is this sort of analysis that enables us to send astronauts to the moon and other such daring feats. The analysis involved in that extraterrestrial situation is much more complicated of course, but the tools and vocabulary that we are now developing in our study of calculus are the fundamental skills that astronauts bet their lives on. It really works.

Rational Functions.

A rational function is one that is constructed as a quotient of polynomial functions. Thus, a rational function is any function that can be expressed in the form

r(x)

p(x)

q(x)

where p and q are polynomial functions. Characteristic features that distinguish rational functions from polynomials include the presence of vertical and horizontal asymptotes in the graphs of some rational functions. You can consult the Field Guide to Functions for the basic concepts, notation and terminology of rational functions.

Rational functions can be differentiated using the quotient rule. The derivative of a rational function is a rational function.

/Stage6/Lesson/algDeriv.html