Step by step
The question
Question: find dy/dx, when x = 2t and y = t².
x = 2t
y = t2
1 dx/dt and dy/dt
The power rule, with t: dx/dt = 2.
dxdt = 2
And dy/dt = 2t.
dydt = 2t
2 dy/dx
The slope dy over dx is dy over dt, divided by dx over dt: the dt's cancel.
dydx = dydtdxdt
So dy/dx is 2t/2.
dydx = 2t2
Simplify: divide the numbers, take away the powers of t.
= t
Careful!
dy goes on top: dy over dt, divided by dx over dt.
Not the other way round.
Careful: not dx/dtdy/dt ✗
So dy/dx = t.
dydx = t
3 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 1: the slope 1 our rule gives.
Right!
t = 1 ± 0.001: ΔyΔx ≈ 1
Let's look back.
First: dx/dt and dy/dt.
Then: dy/dx.
Last: check.
Answer
dydx = t