Intro to Derivatives

Tangent lines are often described as lines that touch a function or curve at exactly one point. This works well for some shapes such as the circle (seen below) but works less well for more complex functions such as the cubic $f(x)=x^3+2x^2+2$ (also seen below).


We're going to need a more refined definition than simple "touching at just one point." To follow the lead of James Stewart - Calculus lets investigate the tangent line for the function $y=x^2$. A tangent line is shown below by the green line, the line is tangent at the point A. The second line goes through the point A and B.

Use the slider to change the x-coordinate of the point B. Notice that as B gets closer and closer to point A the red line gets closer and closer to tangent at the point A.

Lets zoom in a little bit:


If we imagine that point B is getting closer and closer to point A then line connecting A and B will be better and better approximation of the tangent ling at point A. The slope of the line $\overline{AB}$ can be written as:

\begin{align} slope=\frac{\Delta y}{h} \end{align}

The height $\Delta y$ can be expressed as the difference between the y-coordinate of point A and point B. The y-coordinate of point B can be written as $f(x)$ since point B is a point on the function. With that we can write the y-coordinate of point A as $f(x+h)$ since point A is also a point on the function. Thus the slope can be written as:

\begin{align} slope=\frac{f(x+h)-f(x)}{h} \end{align}

As point B gets closer and closer to point A the distance h will shrink getting closer to zero and the slope of the line $\overline{AB}$ will become a better and better approximation of the slope of the tangent line at point A. Take a deep breath and make sure you've got all that.

Notice that the slope of the line is a now a function of x and h. This is a deviation from linear functions where the slope was a constant!

We are on the verge of calculus… and in calculus we often look at the behavior of functions as a variable approaches a particular value. In this case we are interested in the behavior of the slope as h gets closer to zero, that is as the line $\overline{AB}$ gets closer to being a tangent line at point A.

A limit is the value of a function as a variable in that function approaches a particular value. The slope of the tangent line will be the limit of the slope as h goes to zero. This written as:

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

I have labeled this new function $f'(x)$, read as "f prime," both the f and the prime denote that it is somehow related to the original function f. This new function that we have defined is also known as the "derivative" which is the foundation of much of calculus.

The derivative of a function ($f'(x)$) can be viewed as a function that gives the slope of the original function ($f(x)$) at any point x. It turns out this is a critical concept for the development of physics, engineering and more math…

This also allows to create a better definition for a tangent line, which is sort of where we started, a line that is tangent at a point A on a function intersects the function at point A and has the same slope as the function at point A.

Or in terms of the derivative, the slope of a tangent line is equal to the value of the derivative of the function evaluated at the point where the tangent line intersects the function. That was a mouthful…

Note on Notation

The derivative of a function or equation is most often expressed in one of two different notations. The first, which was already used, is written as f-prime such as:

\begin{align} derivative: \: \: f'(x) \end{align}

A derivative comes from the idea of slope and is reflected in second most common notation:

\begin{align} derviative: \: \: \frac{dy}{dx} \end{align}

This second notation is read as "dy dx." This can make a bit of sense if you think of slope as $\frac{\Delta y}{\Delta x}$ where the "delta" is loosely a Greek "d." This notation also has the advantage of directly indicating which variable the derivative is taken with respect to. In the case above the notation "says" that we take the derivative of y with respect to x.

The second notation also suggests some "bad math" by its very nature. It appears to be a fraction, and its not really… In my normal world of physics it can often be treated as a fraction with no harm (but with some foulness). The same notation can also be used without the y to indicate a derivative operator - but that's getting way ahead of the SL syllabus.

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