Step by step
The question
Question: find the slope at t = 1, when x = 3t and y = t² − 4.
Both x and y move with a third letter, t, like time.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 dx/dt and dy/dt
The power rule, with t: dx/dt = 3.
And dy/dt = 2t.
2 dy/dx
The slope dy over dx is dy over dt, divided by dx over dt: the dt's cancel.
So dy/dx is 2t/3.
Simplify: divide the numbers, take away the powers of t.
3 At the point
Put in t = 1: dy/dt is 2 and dx/dt is 3.
The slope is 2/3.
Careful!
dy goes on top: dy over dt, divided by dx over dt.
Upside down gives 3/2: wrong.
So the slope at t = 1 is 2/3.
Draw the curve: for each t, the point (x, y).
The point at t = 1.
4 Check
Let's check with a calculator: move t a tiny bit around 1, and divide the change in y by the change in x.
About 0.667: the slope 0.667 our rule gives.
Right!
5 Pick the choice
Our answer is slope = 2/3.
That is choice C.
6 Why not the others
A is 3/2.
Wrong: dx/dt over dy/dt: it is dy/dt over dx/dt.
That is the Careful mistake from before.
B is 2.
Wrong: used dy/dt only.
D is −3.
Wrong: that is y at t = 1.
Let's look back.
First: dx/dt and dy/dt.
Then: dy/dx.
Then: at the point.
Then: check.
Then: pick the choice.
Last: why not the others.