Question

FULL ANSWERS WITH EVERY STEP PLEASE (10 pts)

Answer 1

Part One
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(f(g))x = 3(7x - 12 + 4/x) - 6sqrt(7x - 12 + 4/x ) + 6 Here I have put g(x) into f(x)
Now you want to differentiate this thing. As much as you can remove the brackets.
21x - 36 + 12/x - 6*sqrt(7x - 12 + 4/x) + 6
Now differentiate.
f(g(x))' = 12 - 12/x^2 + 6 (7 - 4/x^2) / sqrt (7x - 12 + 4/x) 

How come there is now a plus in front of the 6?

Part Two
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I can find no interpretation for the second part other than the obvious. It's been 57 years since I was in a classroom as a student, and any special interpretation to these questions was not discussed.
I take it to mean F(x) / G(x) and then differentiated. If that is not correct, then someone else will have to answer for you. Sorry.

So the question will look like this.
y = [3x - 6(x)^(1/2) + 6] / [7x - 12 - 4/x]
The quotient rule for f(x)/g(x) = [f ' * g - f * g' ] / g^2
 So you need to differentiate f to f ' and g to g'

f ' = 2 + 6/2(x ^ (-1/2 ))

g' = 7 + 4/x^2

Now you can put them in for f and g and f ' and g' to get the answer. It's a lot of typing. Be careful how you put this in. Correct any typos before you send it.

The domain in any question is any value for x except x = 0