Definition of the Derivative

This section covers:

The derivative of a function is just the slope or rate of change of that function at that point.  The reason we have to say “at that point” is because, unless a function is a line, a function will have many different slopes, depending on where you are on that function.

Why would we need to take a derivative in the real world?  Let’s say an object was traveling along a curve, and we wanted to know how fast it was traveling (velocity) at certain points along that curve.  If we had a function for the position of the object at certain times, we could take a derivative at certain points to know the velocity at that time.   So velocity is the rate of change or slope of position.  By the same token, acceleration is the rate of change or slope of velocity.

In fact, calculus grew from some problems that European mathematicians were working on during the seventeenth century:  general slope, or tangent line problems, velocity and acceleration problems, minimum and maximum problems, and area problems.

The reason we need to know about limits is because when we’re dealing with a curve, the actual slope of a part of the curve is constantly changing so theoretically we can’t actually take a derivative.  So we’ll zoom in on that part of the curve and take a use a limit to get the closest we can to the actual slope.

Tangent Line

To illustrate how we take slopes of curves, let’s draw a curve and illustrate the tangent line, which is a line that touches a curve at a certain (only one) point, and typically doesn’t go through that curve close to that point.

However, to get an actual slope of a line, we need two points instead of just one point.  So we must use what we call the secant line to define the slope (average rate of change), where this line goes through two other points on the curve.  But we want this line to be tiny (so the slope is more accurate), so we want to use a limit where the change in x gets closer and closer to 0.

Here are some illustrations.  Do you see how as we get smaller and smaller x values, there’s a much better chance the secant gets closer and closer to the actual tangent (slope) of the curve at points along the curve?  Do you also see that as we get closer, the actual tangent line and secant lines become more and more parallel?  This is what we want when we take the derivative in calculus: the tangent and secant lines basically become the same thing.

Tangent Line Illustrations

Definition of the Derivative

So here is the “official” definition of a derivative (slope of a curve at a certain point),  where  \({f}’\)  is a function of x.   This is also called Using the Limit Method to Take the Derivative.

Do you see how this is just basically the slope of a line formula (change of y’s over change of x’s)?

Definition of Derivative

Don’t let this scare you away from Calculus!   It’s really not that bad, and you actually won’t have to use this equation too often in Calculus.

Again, this derivative finds the slope of the tangent line to the graph of f.  It can also be used to find the instantaneous rate of change, (or rate of change) of one variable compared to another.  And as the  \(x+\Delta x\)  gets closer and closer to 0, the average rate of change becomes the instantaneous rate of change.

Note that not every function is differentiable, especially at certain points; for example, a function might be differentiable on an interval (a, b), but not at other points on its graph.

To use this formula, we usually have to use the Limit Process that we learned about in the Limits section.  The main thing we have to do is eliminate the  \(\Delta x\)  from the denominator since we can’t divide by 0.

And just remember that for  \(f\left( {x+\Delta x} \right)\),  we just put  \(x+\Delta x\)  everywhere where we have an x in the original function. (note that I like to use “h” instead of  \(”\Delta x”\)  since the algebra looks a little less messy).

Here are some examples:

Definition of the Derivative Problems

Here are a few more that are a little more complicated.  Note that sometimes we have to find common denominators, and sometimes we have to use the trick where we rationalize the numerator by multiplying by a fraction with the conjugate on the top and bottom.  The last problem uses trig identities; note that there are other ways to do this using trig identities, but I found this is one of the simplest.

Definition of the Derivative Trig

Equation of a Tangent Line

Note that there are more examples of finding the equation of a tangent line here in the Equation of a Tangent Line section.

Now that we know how to take the derivative (the more difficult way, at this point), we can also get the equation of the line that is tangent to a function at a certain point.  This is because once we know the slope (derivative) of the curve at that point, we have a slope of a line, and a point on that line, so we can get the equation for the line.

So when we get the derivative of a function, we’ll use the x value of the point given to get the actual slope at that point.   Then we’ll use the y value of the point to get the complete line, using either the point-slope or slope-intercept method.   It’s really not too bad!

Here are some examples.  And I promise, taking the derivative will get easier when we learn all the tricks!

Note that in the last problem, we are given a line parallel to the tangent line, so we need to work backwards to find the point of tangency, and then find the equation of the tangent line.

Equation of Tangent Lines

Sometimes we want to know at what point(s) a function has either a horizontal or vertical tangent line (if they exist).   For a horizontal tangent line (0 slope), we want to get the derivative, set it to 0 (or set the numerator to 0), get the x value, and then use the original function to get the y value; we then have the point.

For a vertical tangent line (undefined slope), we want to get the derivative, set the bottom or denominator to 0, get the x value, and then use the original function to get the y value; we then have the point.

Here is an example:

Horizontal and Vertical Tangent Lines

Definition of Derivative at a Point (Alternative Form of the Derivative)

If a derivative does exist at a certain point c (remember that it may not always), then we actually have an “easier” formula for this derivative (slope at this point).  The cool thing is that again this looks just like a slope formula: change of y’s over the change of x’s:

Alternative Form of the Derivative

Let’s do some problems where we use this formula:

Alternative Form of the Derivative Example

Derivative Feature on a Graphing Calculator

You can use the nDeriv( (derivative) function in your graphing calculator to get the derivative (slope) of a function at a certain point;  nDeriv can be found by hitting MATH and then scrolling down to nDeriv( or hitting 8.

Put x in in the denominator (after d, for dx) and put your value for c in at the end (x = c).  Let’s get the derivative on a calculator for the first function above  \(f\left( x \right)=3{{x}^{3}}-1\)  at c = 4, as shown in the first display below.

You can even graph the derivative of a function by using nDeriv  (put x = x at the end) in the “Y = ”  feature.  (Note that the derivative of a cubic function appears to be a quadratic!):


Once you graph a function, you can also use 2nd Trace (Calc) 6 (dy/dx)  to find the derivative of that function at a certain point c.  Once you hit 6 and ENTER, you type in c immediately (even though it doesn’t ask you for it; it will then say X = what you type); in our case, 4.  We see that the derivative at that point is 144 again (you can ignore the Y value):

Derivative with Graphing Calculator

Determining Differentiability

We learned above that not every function is differentiable at certain points.  In fact, the function may be continuous at that point, but not differentiable.   (Note that the converse is true: if a function is differentiable at a point, it is also continuous at that point).

Here are some of the reasons that a function may not be differential at a point x = c:

Not Differentiable

Derivatives from the Left and the Right

We can see that sometimes the derivative is different from the left and the right; in these cases, the function is not differentiable at the point where these derivatives are different.

Here is an example:

Definition of Derivative from Left and Right Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Function Rules  – you are ready!