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: