What actually IS 'differentiation'?

Differentiation is a really common technique in a branch of maths called calculus. The idea is that you have a line on a graph (or a 'function') and we want to know the gradient - the steepness - of that curve at any point along it. This of course implies that we are looking for another function (i.e. we want to be able to put a coordinate - some point along our curve - into this new function, and we want it to give us the gradient of the curve at that particular point). Now the way we find that new function (known as the derivative of our original function) is by a general rule for differentiation: if f(x) = axⁿ, then f'(x) = df/dx = anx^(n-1) . Depending on the focus of the session I would explain differentiation from first principles, but this is just the formula for solving problems. So, our original curve (or 'function') is f(x) and our derivative is df/dx, which can also be written as f'(x). This new function works just the same as the old one, but this time, when we put in some value of x (corresponding to a certain point on the line) it will give us the gradient of the line at that point.