Applying the product rule first you realize you'll be using the chain rule automatically when calculating the derivatives: Take the derivative of the first term,expression,etc and multiply it by the second term. Thats the first part of your product rule. Then ADD: the derivative of the second term multiplied by the first term. (Or whichever order you prefer)
(f*g)' = f'*g + g'*f
The ' , just means the derivative. f' means the derivative of f, and g' means the derivative of g.
Now for the chain rule. When you take the derivative of the first term you notice you need to take the derivative of (x-2)^3, bring the 3 down infront, minus 1 from the exponent, and multiply by the derivative of the inside of (x-2), which is 1:
3(x-2)^2 * 1
A similar operation will be done for the next derivative, if you have problems seeing the "power rule/chain rule" with the square root, just change the squareroot to the power of 1/2. The chain rule is used when you have functions nested inside other functions. So it need not be a power. Say you have cos(5x), the 5x is nested inside the cos(). So taking the derivative you would get -sin(5x)*5. Generally it's just a good idea to remember the power rule and the chain rule applied to trigonomic functions. If you need more examples or maybe another way of describing it, just google power rule or chain rule examples or search the forums. If you really want to know where it comes from, try picking up an introductory analysis textbook or just search for a proof.
Quick example, try doing this one first. Its a little easier: The derivative of:
(5x-1)^2 (4+2x)^3<br />
=(5x-1)^2*3(4+2x)^2 2 + (4+2x)^3 *2(5x-1) 5