If #f(x) =-sqrt(2x-1) # and #g(x) = (2-1/x)^2 #, what is #f'(g(x)) #?

Answer 1
The question is asking, "what is the equation for the derivative of #f#, when its input is #g(x)#?
So we need to do two things: (1) Find the general derivative of #f#, and (2) Plug in #g(x)# as the input to this derivative.
Step (1): If #f(x)=-sqrt(2x-1)=-(2x-1)^(1//2)#, then #f'(x)=-1/2(2x-1)^(-1//2)(2)# #=>f'(x)=-1/(sqrt(2x-1))# by use of both the power rule and the chain rule.
This is the general derivative of #f#, with respect to some input #x#. If we plug in different inputs, we get different outputs.
Step (2): Now, we plug in the input #g(x)# (instead of just #x#) to the derivative #f':# #f'(color(red)x)=-1/(sqrt(2color(red)x-1))#
#=>f'[color(blue)g(x)]=-1/(sqrt(2color(blue)((2-1/x)^2)-1))#
#=>f'[g(x)]=-1/(sqrt(2(2-1/x)^2-1))#.

Depending on your teacher, this is likely as far as you'll need to go.

Please note: The domain of #f'(x)=-1/(sqrt(2x-1))# is #{x|2x-1>0}#, which is #x>1/2#. (Slightly more limiting than the domain of just #f(x)#, which is #x>=1/2#)
The domain of #g(x)=(2-1/x)^2# is #{x|x!=0}#.
That means the domain of this composite function #f'[g(x)]# has to satisfy both of these restrictions. Luckily, that's pretty easy, considering the domain of #f'(x)# is a subset of the domain of #g(x)#. So the domain of #f'[g(x)]# is #{x|x>1/2nnx!=0}#, or just #{x|x>1/2}#.
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Answer 2

To find ( f'(g(x)) ), you need to find the derivative of ( f ) with respect to ( x ) and then substitute ( g(x) ) in place of ( x ).

First, find ( f'(x) ): [ f'(x) = \frac{d}{dx}(-\sqrt{2x-1}) ] [ f'(x) = -\frac{1}{2\sqrt{2x-1}} ]

Now, substitute ( g(x) ) into ( f'(x) ): [ f'(g(x)) = -\frac{1}{2\sqrt{2g(x)-1}} ]

Substitute ( g(x) = (2 - \frac{1}{x})^2 ): [ f'(g(x)) = -\frac{1}{2\sqrt{2(2 - \frac{1}{x})^2 - 1}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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