The technical differentiation of functions required for the TMUA is actually fairly basic, with only rational powers of x being included on the specification. However, what is required, is a good understanding of what differentiation means; both generally, as a rate of change and as the gradient of the tangent to a graph. Questions often combine differentiation with another topic, such as Coordinate Geometry or Graphs of Functions.

The product and quotient rules are not in the specification, so any product or quotient can be expanded and simplified to give a sum of powers of x before differentiating. However, if you are confident with using the product or quotient rule, then you can obviously use that if you prefer. The simplification required links back to the indices and surds topic in 1.1 and 1.2 (Algebra and Functions).
 

Some questions will be similar to standard A Level questions about finding the equations of tangents or normals, or might extend this to finding distances between curves and lines, (see Coordinate Geometry for practice). 
 

By differentiating a function and substituting different x values, we can calculate the value of the gradient of the tangent to the graph at these points, and tell if the function has a stationary point, or if it is increasing (positive gradient) or decreasing (negative gradient). By considering the position of these points (by calculating the y values of the original function), we get further information about the shape of the graph and possibly the number of roots. By finding the second derivative we can further determine whether the graph is convex or concave for different domains, and whether a stationary point is a maximum or a minimum. All of this can help us to sketch functions such as higher order polynomials.