This section covers:

**Tangent Line****Definition of the Derivative****Equation of a Tangent Line****Definition of Derivative at a Point (Alternative Form of the Derivative)****Derivative Feature on a Graphing Calculator****Determining Differentiability****Derivatives from the Left and the Right****More Practice**

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. We’ll zoom in on that part of the curve and 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. 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.

# 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)?

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:

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.

# 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

**value of the point to get the complete line, using either the**

*y***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.

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:

# 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

**’s over the change of**

*y***’s:**

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

# 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 **2 ^{nd} 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):

# 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*:

# 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:

**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!