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When we find dy/dx we find the gradient of the curve at (3,1). Start by differentiating the left hand side (LHS) like so.. (whiteboard). Remember every time we differentiate a y value we multiply by dy/dx...

2(x+3)/(x+1)

One can treat complex multiplication as polynomial multiplication, but remembering i^2 = -1. To perform polynomial multiplication, multiply each term one by one, then add them together. Hence (3+i)(1+2i) ...

This question is a typical example aimed to test the student's understanding of the quotient rule, a technique which is used very often in calculus problems. Answer: For a function f(x) = cosx/sinx = u/v,...

We can identify xe^(-2x) as a product and hence we will most probably need to use integration by parts.

We then set u = x and v' = x^(-2x). It is important to do it this way round so that when we d...