x = t^3, y = t^2, find dy/dx

Step by step

The question

Question: find dy/dx, when x = t³ and y = t².

1 dx/dt and dy/dt

The power rule, with t: dx/dt = 3t².

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/(3t²).

Simplify: divide the numbers, take away the powers of t.

Careful!

dy goes on top: dy over dt, divided by dx over dt.

Not the other way round.

So dy/dx = 2/(3t).

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 0.667: the slope 0.667 our rule gives.

Right!

Let's look back.

First: dx/dt and dy/dt.

Then: dy/dx.

Last: check.