What is the first derivative of y=5z(1+2z2)? Is this a minimum, maximum or turning point?

You should not be put off by the algebraic letters used, substituting x's for z's is absolutely fine. For the first part of the question, since the equation involves the product of two terms, there are two ways one could begin to work out the first derivative i.e. dy/dz. The first is to use the chain rule. In this instance, however, it is probably easier to multiply out the brackets and then use simple differentiation. The answer is dy/dz = 5+4z.

To find out whether this derivative is a minimum, maximum or turning point we take the derivative of the derivate which equals 4. Since 4 >0, this point is a minimum. We can check this visually by plotting the original equation as a graph.