I hoped you've managed to solve these questions. Now, it’s our chance to check how you've done.
So, let’s go through them together. Our first example is giving us two equations. It’s asking us
to differentiate. Obviously, it’s important that you recognize that this is a parametric equation.
So, we’ll have to differentiate this parametrically. Let’s have a look at our two equations.
We have y = 3t + 5 and then we have x = 3t - 5. The rules here are straightforward.
You differentiate each one of them separately, so dy/dt and dx/dt and then you bring them together
to give you dy/dx. So, dy/dt gives you 3. We’re differentiating with respect to t, so bring the power down
and decrease the power by 1. Dx/dt also gives me 3. Now remember that when I used my definition
dy/dx, that is the same as dy/dt multiplied by dt/dx. So, I want to flip this. Rather than dx/dt,
I want to write this as dt/dx which is just 1 over 3. Now, when I put it all here, I’ll get dy/dt
which is 3 and dt/dx which is 1 over 3 giving me a nice, easy gradient of just 1.
This is fairly straightforward. It’s just telling us the steepness of this curve is just 1 at this point.