Derivatives by First Principle

Ever so rarely the IB asks a question that requires the students to find a derivative from "First Principles." Many if not most math classes start calculus with something closely related to first principles. I don't and thus I present it as a separate topic.

Comment to Math Teachers: Feel free to flame or troll all you want, this is 2012 and their are better ways to introduce Calculus than first principles. I had the technology advantage of Lego Robots and Logger Pro to start calculus which made all the difference. I'm sure similar things could be done with a simple battery operated car and some simple circuits at a fraction the cost of Lego Robots.

#### First Principles

Essentially first principles is a fancy math term (used in physics too) to say we go back to the most basic definition. Well, the derivative at its base level is about slope.

So we would say that "as h approaches zero" then the slope of the function is the derivative. Or in math symbols:

(4)
\begin{align} f'(x) =\lim_{h \rightarrow 0} \frac {f(x+h)-f(x)}{h} \end{align}

#### Examples

The nice thing about these types of problems in SL math is you should generally know the answer, but the hard part is showing the work!

Example 1
Find the derivative of $f(x)=x$ using first principles.

Example 2
Find the derivative of $g(x)=x^2$ using first principles.

#### Basic Outline of How to Solve Problems

1. Write the definition of the derivative
2. Use the definition of the function to subsitute into the definition of the derivative
3. Expand (and then simplify) the algebra
4. With polynomials you will always be able to "divide by h"
5. Ask yourself what would happen the the expression as h got really small? The result is your answer.
6. Check your result using basic derivative rules

#### A few comments on IB Problems

The examples done above are a bit simpler than the average IB "First Principles" problem, but only in the messiness of the algebra. Its not uncommon to have to find the derivative of a cubic function by first principles. Which results in having to expand a term that looks like $(x+h)^3$. You many even get a fourth order function (quartic) so you have to expand $(x+h)^4$. In both cases, or heaven forbid a 5th order, using a binomial expansion is faster and easier than grinding through "FOILing" or whatever other method you learned. These two topics are often put into a single problem.

Also in SL you ONLY have to find derivatives of polynomials by first principles. You HL'ers… well, sorry, it's HL.