Question on differentiation?

This is going to be tricky to demonstrate on here without mathematical symbols, but I'll try my best:

For this description I will use the following notation:

* means 'multiplied by'

/ means 'divided by'

^ means 'to the power of'

pi means 3.1415926535...

A = 4pi r^2

where A is the surface area of the bubble and r is the bubble's radius. This could be re-written:

dA = 4pi dr^2

Now, we know that:

dr/dt = 0.2

We can approximate

dr^2/dt = 0.2r

From the equation above (multiplying both sides by dr). Now rearrange the second equation to get:

(1/4pi) * dA = dr^2

and divide through by dt:

(1/4pi) * (dA/dt) = dr^2/dt

We already know dr^2/dt from the fourth equation above, so we can rewrite this as:

(1/4pi) * (dA/dt) = 0.2r

And rearranging this gives:

dA/dt = 0.2 * 4pi * r

We want to find dA/dt at the radius 7cm, so this is:

dA/dt = 0.2 * 4pi * 7 = 17.6 cm^2/s

And that's the answer, 17.6cm^2/s

PS: I think I have this right but I've been off for Christmas and my brain has turned itself off. The units all balance and I can't see any obvious mistakes but if anyone sees any, please say!!!

PPS: Ojo's differentiation is with respect to the radius, not with respect to time (as is required)

Differentiation can be interpreted as the rate of change of area.

We need to get the differential of 4pir^2

the differential of 4pi r^2 is 8pi r dr

8 pi (7cm)(0.2cm/s)= the answer is 11.2cm^2/s