Let’s delve into the “exciting” world of Calculus, starting with **Differential Calculus**! Think of it this way: **Algebra helps us find the slopes of lines, and Differential Calculus helps us find the slopes of curves.**

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. **Velocity**, then, 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 **$ \boldsymbol{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? 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

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)? Yes, $ \Delta x$ means “change in $ x$”, but for now, think of it as another variable.

Note that just the quotient part of this formula (without the **lim**, or limit) is called the **Differential Quotient**. 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. And 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. For example, polynomials are typically differentiable, but rational functions are not at certain points (because of removable discontinuities and/or asymptotes).

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 just **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**.

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 the variable “$ 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** (changing the sign between terms) 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. **Trust me; there are easier ways to take these derivatives!**

## Equation of a Tangent Line

Note that there are more examples of finding the equation of a tangent line (including **horizontal and vertical tangent lines**) 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.

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** $ (y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right))$ or **Slope-Intercept** $ (y=mx+b)$ method (for me, preferred). It’s really not too bad! (Weird fact: the equation of a tangent line for a linear function is just that function!)

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.

## 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:

Here are some problems where we use this formula:

## Derivative Feature on a Graphing Calculator

You can use the **nDeriv(** (derivative) function on the **TI graphing calculator** to get the derivative (slope) of a function at a certain point; hit **math** and then scroll down to **nDeriv(** or hit **8**. To get the derivative at a certain point, put $ x$ in the denominator (after $ d$, for $ dx$) and put the value for $ c$ in at the end ($ x=c$). You can even **graph the derivative** of a function by using **nDeriv **(put $ x=x$ at the end) in the **Y =** feature. Here are examples for the derivative of $ f\left( x \right)=3{{x}^{3}}-1$ at $ c=4$.

You can also input a function and find the derivative at any point. After inputting and graphing the function (making sure the $ c/x$-value is in the window), use **2 ^{nd} trace** (

**calc**)

**6**($ dy/dx$), hit

**ENTER**, and 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).

You can also put a function in **Y _{1}** and put the derivative in

**Y**by using

_{2}**nDeriv(**with $ x$ after the “$ d$”,

**alpha trace enter**(

**Y**) for the function, and then $ x$ at the end. Then you can see a function and its derivative on the same graph. Note that the derivative of a cubic function appears to be a quadratic.

_{1}## Determining Differentiability

We learned above that **not every function is differentiable at certain points **(for examples, polynomials are differentiable at all points, while rational functions are not). In fact, the function may be continuous at a certain 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.

**Thus, in order for a function to be differentiable at a point **$ \boldsymbol {x=c}$**, **$ \boldsymbol {f\left( x \right)}$** must be continuous at **$ \boldsymbol {x=c}$**, and **$ \displaystyle \boldsymbol {\underset{{x\to {{c}^{-}}}}{\mathop{{\lim }}}\,{f}’\left( x \right)=\underset{{x\to {{c}^{+}}}}{\mathop{{\lim }}}\,{f}’\left( x \right)}$**.**

Here is an example:

Here are a few types of **Piecewise Function** problems you might see to make sure you understand if functions are continuous and/or differentiable. Note that we used shortcuts finding the derivatives that you’ll soon use in the **Basic Differentiation Rules** section.

**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!