Step by step
The question
Question: find the slope at t = 2, when x = t³ and y = 6t².
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 = 3t².
And dy/dt = 12t.
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 12t/(3t²).
Simplify: divide the numbers, take away the powers of t.
3 At the point
Put in t = 2: dy/dt is 24 and dx/dt is 12.
The slope is 2.
Careful!
dy goes on top: dy over dt, divided by dx over dt.
Upside down gives 1/2: wrong.
So the slope at t = 2 is 2.
Draw the curve: for each t, the point (x, y).
The point at t = 2.
4 Check
Let's check with a calculator: move t a tiny bit around 2, and divide the change in y by the change in x.
About 2: the slope 2 our rule gives.
Right!
5 Pick the choice
Our answer is slope = 2.
That is choice D.
6 Why not the others
A is 1/2.
Wrong: dx/dt over dy/dt: it is dy/dt over dx/dt.
That is the Careful mistake from before.
B is 24.
Wrong: used dy/dt only.
C is 4.
Wrong: 4/t: put in t = 2, not 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.