Use the chain rule to show that, if y = sec(x), then dy/dx = sec(x)tan(x).
First, write y in terms of cos(x). We are familiar with cos(x) and know how to differentiate it. We know that sec(x) = 1/cos(x) = (cos(x))-1. Next, find dy/dx in terms of cos(x) and sin(x). Again, we are familiar with cos(x) and sin(x) and will be able to get the answer in terms of sec(x) and tan(x) later. Time to use the chain rule. Multiply by the current power, reduce the current power by 1 and multiply by the differential of what is inside the brackets. Hence, we end up with: dy/dx = -1 x (cos(x))-2 x -sin(x) = sin(x)/cos2 (x) because the minus signs cancel each other. Finally, we know that we need to get dy/dx in terms of sec(x) and tan(x), so we need to look for a way to achieve this. We can get this by separating sin(x)/cos2 (x) into sin(x)/cos(x) x 1/cos(x). Using our knowledge of the definitions of sec(x) and tan(x), we can see that this is just sec(x) x tan(x), as required.
