x = t^3, y = 6t^2, find the slope at t = 2

Step by step

The question

Question: find the slope at t = 2, when x = t³ and y = 6t².

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.

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!

Let's look back.

First: dx/dt and dy/dt.

Then: dy/dx.

Then: at the point.

Last: check.