Using Differentiation to Find The Slope of a Tangent

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,

“... they went off on a tangent ...”

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!

SEE IMG

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:

Rise
Run

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).

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,

function f(x) shown with point P marked alongside its tangent

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,

function f(x) shown with point P marked alongside its tangent and a close-by point, T also shown with its tangent

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 =

mPT =

Rise
Run
  =
Change of y
Change of x
  =
f(b) - f(a)
b - a
  =

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.

Theoretical Definition of the Slope of a Tangent

mTan = lim[b -> a] 
f(b) - f(a)
b - a

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.