## Guest Post: Derivatives and an Introduction To Calculus

by Shaun Klassen

One of the mathematics subjects most feared by students is the “dreaded” differential calculus subject.  Absolutely, it is more complicated than more common basic algebra that most would have studied up to this point.  And of course, to work with calculus, one must be familiar with all of the earlier concepts that build up a strong mathematical foundation, including things like algebra, trigonometry, and graphing.  However, this is not to say that calculus has to be hard, or “impossible.”  It is completely doable if you start slowly by learning the general problem solving strategies.  In this guest post, I want to introduce the main concept of differential calculus – the derivative – and I encourage you to visit my math website to find out much more information about this subject.

The name “differential calculus” is a descriptive one – it is based on differences, or changes.  More specifically, it is all about describing how one quantity changes with respect to another one, or in other words, the rate of change.  The derivative is used to express this function, but let’s examine this concept a little more closely by considering everyone’s favourite rate of change: velocity.  » Read more

## The Algebraic and Geometric Meaning of Derivative

Note: This is the first part of the Derivative Concept Series. The second part is Derivative in Real Life Context and the third part is Derivative and the Maximum Area Problem.

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If we want to get the slope of a line, we need two points. Suppose the points have coordinates $(x_1,y_1)$ and $(x_2,y_2)$, we have learned that the slope is described by the formula $\displaystyle\frac{y_2-y_1}{x_2-x_1}$.

In Figure 1, we have line $l$ tangent to the function $f$ at point $P$ where the coordinates of $P$ are $(x,f(x))$.  The problem that gave birth to calculus is getting the slope of this tangent line. There is, however, a problem.  We need two points to compute for the slope but we have only one point.

Note that the word tangent in this problem is different from the definition of tangent on a circle because it is clear that line $l$ will intersect the graph in more than one point.

Figure 1 - Line l tangent to the function f at point P.

Using the concept of limits we can remedy this problem. First, we create point $Q$ with x coordinate $h$ units to the right of the x-coordinate of $P$. We then draw line $PQ$, a secant line to the function $f$.

Figure 2 - A secant line is drawn through P.

In effect, the coordinates of $Q$ would be $(x+h, f(x+h))$ and it is clear that the slope of the secant line $PQ$ is described by the formula

$\displaystyle\frac{f(x+h)-f(x)}{x+h - x} = \frac{f(x+h)-f(x)}{h}$

If we want to approximate the slope of the tangent line, it is reasonable that we move $Q$ towards $P$ with $P$ fixed. Click here to explore the diagram above using GeoGebra.

From the GeoGebra exploration above, if we move $Q$ towards $P$, we observe the following:

1.)    The value of $h$ approaches $0$.

2.)    The inclination of the secant line approaches the inclination of the tangent line.

3.)    The slope of the secant line approaches the slope of the tangent line.

4.)    If point $Q$ coincides with point $P$, then the slope of the secant line and is equal to the slope of the tangent line.

If we let $m$ be the slope of the secant line and$f'(x)$ be the slope of the tangent line, focusing on observations 1 and 4, we can say the following equivalent statements:

• The limit of the slope of the secant line $m$ as $Q$ approaches $P$ is equal to $f'(x)$.
• The limit of the slope of the secant line $m$ as $h$ approaches $0$ is equal to $f'(x)$.
• The limit of $\displaystyle \frac{f(x+h)-f(x)}{h}$ as $h$ approaches $0$ is equal to $f'(x)$.

Using the limit notation, we can say that

$f'(x) = \lim_{h \to 0} \displaystyle\frac{f(x+h)-f(x)}{h}$

From the above discussion, we can see that the derivative of a function at  a particular point is the slope of the line tangent to that function at that particular point.

In the next post, we will discuss the meaning of derivative in real life situations.