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

Step by step

The question

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

1 dx/dt and dy/dt

The power rule, with t: dx/dt = 2t.

And dy/dt = 3t².

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 3t² over (2t).

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 = 3/2 × 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 1.5: the slope 1.5 our rule gives.

Right!

Let's look back.

First: dx/dt and dy/dt.

Then: dy/dx.

Last: check.