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Calculus I: The Derivative

Welcome to the first lesson on CalculusTricks.com!

Things you need to know before we start the lesson

  • Tangent Line – A straight line that just barely skims across the surface of a curve. Locally, it touches at exactly one point, but it might cross again at other parts of the graph.
  • Secant Line – Another straight line that crosses through a curve. It can hit multiple points of the curve.
  • Slope – If you know two points of a line, (x1, y1) and (x2, y2), you can calculate the slope of that line using the formula . It doesn’t matter which point is 1 or 2, if you switch them you will get the same answer.
  • Limits – If you have questions on limits, I’ll be posting a lesson on them in the near future!

What is the derivative?

Okay, so let’s just start with a tiny bit of history. Mathematics is never created for no reason. Scientists needed ways to figure things out, and math is just the tool to do so. So, the derivative was created to solve a specific problem. We needed to be able to find the exact slope of a line that is tangent to a curve. Well, the best way to start is with a good approximation.

In the image below, I have drawn a random curve and three straight lines.

The black line is tangent to the curve at the point (x, f(x)). The red line is a secant line. This line runs through the points (x, f(x)) and (x+h, f(x+h)). h is a very small change of x. So x+h is just a little bit to the right of x. If we found the slope of this line, it would be a pretty good approximation for the slope of the tangent line.

    the x’s cancel in the denominator!

But instead of doing that, what if we made h a little bit smaller? The green line is also a secant line, but now h is a little bit smaller. Wouldn’t this be a better approximation? What if we made h even smaller than that? The equation for the slope would be the same, but the h would be smaller and smaller. We can’t let h=0, but what if we made it infinitely small? What if we took the limit as h goes to 0? Wouldn’t that be the exact slope of the line?

The formula above is called the definition of derivative. If you see a problem that says, “use the definition of derivative to find the slope of the tangent line,” this is the formula to use!

What is the symbol for the derivative?

The derivative has many symbols. The most common is ‘ or “prime.” If you are given an equation f(x), the derivative is simply written as f ‘(x). This is said out loud as, ” f prime of x.” If you are given an equation y, the derivative is y’ or “y prime.” Other ways that you might see the derivative are  (another very common notation), and . These are both spoken as “the derivative of y with respect to x.” However,  can also be called “dee-why-dee-exx” (just saying the letters).

So, how do we use it?

Simple! Let’s say that we have an equation, . Using the definition of derivative, find the slope of the tangent line at the point x = 2.

Now, you must ask yourself, what is f(2)? We know that . So since 2 is in the parenthesis (instead of x), wherever you see an x, put a 2! .

 Now, what is f(2+h)? We do the same thing as before. . Inside the parenthesis is 2+h, instead of x. Wherever there is an x, wtire 2+h. .

The next step is to substitute f(2) and f(2+h) back into the limit.

Now we know that the slope of the line that is tangent to the curve  at the point x = 2 is 8!

What if we were to think about this a bit more in general? This is how you find the slope of the tangent line at a specific point. What about at any point? What if you are not given a point? Fear not! The answer is simple. All you do is leave the x’s alone, and find the slope the exact same way.

Using the definition of derivative, find the slope of the line that is tangent to the curve .

Slope = 4x – 5. What does this mean? This describes the slope of the line that is tangent to the curve  at any point on the curve! That is a pretty powerful statement! Now, if anyone were to ask us about a specific point, it would be very easy to tell them! What is the slope of the tangent line at the point x = 3? Just plug 3 into 4x-5 = 12-5 = 7!

There are many applications to this type of problem. I’ll show you one below, and then I will post a few more in the example section here!

A common problem you will see is to find the equation of the line tangent to a curve. This kind of problem will always give you the equation of the curve in the form f(x) = __ or y = __  and a point on the curve. The first step to solve this problem is to find the slope of the tangent line. You already are given a point that is on the tangent line. Since you know a point, (x1, y1), and the slope, m, you can easily find the equation of the line using the formula: y = m(x – x1) + y1. Let’s try one!

Find the equation of the line that is tangent to the curve  at the point (1, -2).

This is the equation for the line that is tangent to the curve  at the point (1,-2)!
Simple as that =)

Please give me any feedback that you might have, so that I know how to better help you! Also, check out the next lesson on Basic Derivative Rules.