Derivatives Tutorial

Tutorial Module on Derivatives

Basic Derivatives

The derivative of a function, f(x), at a point can be thought of as the slope of a tangent line at that point. Or, another way of thinking of it is the rate of change of the function at that point. In statistics, we use the derivative to find the rate at which the probability is changing.

The simplest derivative to take is of the function f(x)=c, where c represents a constant. This function is not changing as the variable x changes, therefore the derivative, represented by , will be equal to 0. That is, = 0.

If , then the derivative will be .

For example:

If then

Likewise

If then

If the power on x is negative, the same rule holds:

If then

To see what is happening, take a look at the following graphs,

the one on the left (fig.1) represents
and that on the right (fig.2) represents the derivative

fig.1

fig.2

As x increases in the graph on the left, the rate of change of the function increases (i.e. the curve has a steeper rise). Because of this increase, the derivative consists of an increasing function.

When a function of x is multiplied by a constant, the derivative will be the constant times the derivative of the function.

For example:

if then,

If consists of a sum of functions of x, like , then the derivative is found by taking the derivative of each piece:

The Chain Rule

To find the derivative of , we could multiply the power through and then use the methods above, however that is a lengthy way to go about it and it is easy to make mistakes. A simpler, faster method is to use the chain rule:

If y = u(x) is a differentiable function of u and u(x) is a differentiable function of x, then the derivative of y(u(x)) can be found by taking the derivative of y with respect to u and multiplying by the derivative of u(x) taken with respect to x.

What does this mean? Well, in terms of our example, let y = u⁴ and u = x² - 5 then, .
What I did there was to take the derivative of y with respect to u: (4u³) then the derivative of u with repect to x: 2x multiply the two together and replace the u with x² - 5, then simplify the result.

Another example:

,
then

The derivative of e^x:
A commonly used function in startistics is e^x, so we should cover how to take the derivative of that function.

If then

So if e is raised to a power the derivative is e raised to the power and taken times the derivative of the power.

For example:

gives us the derivative .

Second example:

then, .

Problems to Try:
For each of the following functions on the left, take the derivative. The answer is given on the right.

the derivative is

the derivative is

the derivative is

the derivative is

Summary:
Here is a summary of the derivatives commonly used in statistics, along with another example of each:

Now, try the quiz on derivatives and see if you do better this time. If not, contact Dr. Barnet (barnetb@uwplatt.edu) for further assistance.