Firstly, a tangent is a straight line that touches a curve at a certain point. Imagine you are riding your bicycle on the road around a bend of a park, imagine yourself steering the handlebars to go around the bend. Then, a dog comes up and starts to bark at you at a specific point you are steering around the bend, you straighten your handle bars and then stop following the bend and go off on a straight line, to avoid the dog. That straight line is the tangent to the bend, or for our purposes, then tangent to the curve.
You may have also heard the expression about someone going off topic about something,
Meaning they were once on topic and have now stopped following the topic and gone in a different direction. For example, if we were having a discussion about how to bake a cake, and we were talking about what temperature to set the oven at, and then someone starts to talk about their oven and what type of oven it is and where they bought it from and how their fridge comes from the same company etc…. That is going off on a tangent. Think of that example if it helps!
Now, switching it back to math, just imagine that point mathematically, as coordinates, and imagine that tangent as a straight line. Here, we will learn how to calculate the slope of the tangent, which means the gradient of the tangent. First, we will start with an approximate way to calculate the slope of the tangent.
Remember, the slope of a line (the gradient) can be thought of as:
Or, for a standard y and x axis graph, the change in the y-value over the change in the x-value.
Now, say we have the curve of a function, the function is represented by y = f(x).
Remember, f(x) means a function of x, which basically means an equation to do with x, so y= f(x) could mean y = 3x + 1 or y = x2 - 2x + 1 or any other variation. In those examples, the function would be the 3x + 1 or the x2 - 2x + 1. When it says a function of x or f(x) don't have to concern yourself with what it specifically means.
For this curve y = f(x), we have the point along the curve, P, the x-value of this point is a, so the y-value of that point (subbed into our equation y = f(x)) is f(a).
So, the point P has the coordinates x = a and y = f(a),
Or (a, f(a)),
Now imagine the tangent that touches the point P,
Recall, to find the slope of a line, we subtract the two y-values of two points and divide them by the subtraction of the two x-values of those two points.
For finding the slope of the tangent for the point P in our illustration above, we only know the value of one point on that tangent, as there is only one point where the tangent touches the curve (the curve being y = f(x)). If we take another point along that curve, say T, that is close to the point P, we can work out an approximate value of the tangent, as notice, the line drawn connecting the point P and the other point T looks approximately the same as the line that is the tangent to P.
Say the x-value for the point T is b, so the y-value is f(b) [remember, Q is also on the curve y = f(x)]
Now, lets look at the line between P and T,
We can see that the line between P and T is approximately close to the line that is the Tangent of P (mTan). If the point T was closer to P, the line between them would be getting closer to what the Tangent of P (mTan) is.
Calculating the slope (gradient, m) of the line between P (a, f(a)) and T (b, f(b)), which we will represent by mPT
Slope of PT =
Now, as the point T gets closer to the point P, mPT (gradient of the line PT) gets closer to mTan (the gradient of the tangent of the point P), or
As the x-value of a gets closer to b, the gradient of the tangent becomes closer to the gradient of the line between those two points. Mathematically we write this as.
mTan, or the slope divided by the gradient of the tangent is called the derivative of the function f(x) at the point P.
What the equation above is saying is, as the value of b gets closer to a, then the slope of that line gets closer to the slope of the point a. The limit of how close b can be to a is equal to the slope of the tangent to a.
This might all seem very abstract but as we work through this you will start to understand it.