Finding the Derivative of an Inverse Function

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The discussion centers on finding the derivative of an inverse function, specifically using the example of f(x) = x^2 and its inverse g(x) = x^(1/2). The derivative g'(4) is calculated using the formula g'(y) = 1/f'(x), leading to the conclusion that g'(4) = 1/4. Participants debate the clarity of expressing g'(4) in terms of f'(x) without explicitly stating x, with some suggesting that specifying x is important for accuracy. Overall, the consensus is that while both methods for finding g'(4) are mathematically correct, clarity in communication is essential, especially in academic settings.
UrbanXrisis
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I was taught in class that if g(x) is f^-1(x), then g'(y)=1/f'(x)

my notes for an example:
f(x)=x^2
g(x)=f^-1(x)=x^(.5)
g’(4)=?
g(x)= x^(.5)
g’(x)=.5x^-.5
g’(4)=1/f’(x), x=4^(.5)
g’(4)=1/f'(2)=1/2*2=1/4

I have no clue how g(x)=f^-1(x)=x^(.5) and how g’(4)=1/f'(2)

I am just overall confused any help would be appreciated
 
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f(x) = x^2

set f(x) = y

y = x^2

(*To find inverse solve for x explicitly in terms of y*)

y^0.5 = x

f^-1(x) = x^0.5

let g(x) = f^-1(x)

g(x) = x^0.5

The second one follows the same way
 
I have:
g’(4)=1/f'(2)=1/2*2=1/4

and also:
g'(4)=1/f'(x)=1/f'(2)=1/4

is both ways acceptable?
 
UrbanXrisis said:
I have:
g’(4)=1/f'(2)=1/2*2=1/4

and also:
g'(4)=1/f'(x)=1/f'(2)=1/4

is both ways acceptable?

I wouldn't go with the second one. It could be seen as wrong to say g'(4) = 1/f'(x) without saying what x is explicitly, which in this case is root(4).

But, it depends how picky your instructor is
 
what about:
g’(x)=.5x^-.5=.5(4)^-.5=1/4

could that be acceptable too?
 
UrbanXrisis said:
what about:
g’(x)=.5x^-.5=.5(4)^-.5=1/4

could that be acceptable too?

Sure,

g(x) = x^\frac{1}{2}

thus,

g'(x) = \frac{1}{2x^\frac{1}{2}}

g'(4) = \frac{1}{2*4^\frac{1}{2}}

g'(4) = \frac{1}{2*2}

g'(4) = \frac{1}{4}


Again, I would just recommend that you specify you are changing from the general equation g'(x) to g'(4) somewhere in your solution. But if its not on a test, or your teacher isn't picky then that should be fine.
 
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