The greatest leap forward in modern mathematics was made in the 17th century, when the calculus was developed independently by Isaac Newton in England and Gottfried Leibniz in Germany.

There are many popular books (i.e. without mathematical exercise inside) written on this topic, a good recent one is “Infinite Powers: How Calculus Reveals the Secrets of the Universe” by Steven Strogatz. An even easier, but still very valuable read, if you prefer a very general introduction is David Acheson’s “The Calculus Story: A Mathematical Adventure”.

As far as we are concerned at iGCSE level, there exists a formula that can be used to find a gradient function for a curve. That means that if we apply this formula to a function that we know, the answer will be a gradient function (also called a derivative function) that will tell us the gradient at any point along the curve. This formula is:

The process of using the formula is called differentiation. Let’s try it out with some examples offered to us by the teacher.

Exercise

Let’s complete exercise 17 on page 268 of the extended textbook:

As we mentioned above, the result is the gradient function. If we wish to find the gradient at a certain point (x,y), we simply substitute the value of x into the gradient function and the output of the function dy/dx tells us the gradient. Let’s try this with some examples given by the teacher.

Exercise

Let’s complete exercise 18 on page 269 of the extended textbook:

Turning Point

One use of the gradient function is that it can help us to find turning points (or stationary points) on a graph, because at these points the gradient function is equal to zero (i.e. dy/dx = 0).

By differentiation the gradient function once more (i.e. finding the second derivative) we can also find the nature of those stationary points (are they maximums are minimums or point of inflections). A maximum has a negative second derivative, a minimum has a positive second derivative, and a point of inflection has a zero second derivative.

Exercise

Let’s complete exercise 19 from page 271 of the extended textbook: