Any thoughts @tkhunny
@zepdrix and I tried to solve this one, but it didn't work out @tkhunny
How do you know it didn't work out?
Because -11 is not one of the answer choices
the answer is 1
damn sure
how?
Just one thought. You missed the easiest problem on the exam. You should know this: \(\dfrac{d}{dx}f^{-1}(x) = \dfrac{1}{f'(f^{-1}(x))}\) Rewriting in terms of \(g(x) = f^{-1}(x)\;for\;x=0\) \(\dfrac{d}{dx}g(x) = \dfrac{1}{f'(g(x))}\) Are you seeing it, yet? Rephrasing the LHS, and substituting \(x=0\) in the usual slightly incorrect way \(g'(0) = \dfrac{1}{f'(g(0))}\) Has it hit you, yet? Multiplying my the hopefully nonzero denominator on the RHS \(f'(g(0))g'(0) = 1\) -- Well, THAT looks familiar!
How do you know that \(\dfrac{d}{dx}f^{-1}(x) = \dfrac{1}{f'(f^{-1}(x))}\)?
@tkhunny @phi
A proof of that property should be in your text book
I'm looking for it..
The formula is in the textbook,but there isn't an explaination of why beyond the fact that derivatives of inverse functions are reciprocals of each other
It's somewheres between obvious and totally confusing. let y=f(x) and x= g(y) where g is the inverse function of f for a specific (x0, y0) point we know y0= f(x0) and x0= g(y0) form the fraction \[ \frac{g(y)-g(y_0)}{y-y_0} = \frac{x-x_0}{y-y_0} = \frac{1}{\frac{y-y_0} {x-x_0}}\] now take the limits \[ g'(y_0) = \lim_{y\rightarrow y_0} \frac{g(y)-g(y_0)}{y-y_0} =\frac{1}{ \lim_{x\rightarrow x_0}\frac{y-y_0} {x-x_0}}= \frac{1}{f'(x_0)}\]
I get confused at the limit part
that is the definition of derivative
no, I mean I get taking the limit/derivative, but I don't get what's after the second = sign
that is supposed to be the limit as x->x0 of the fraction we can "move" the limit into the denominator because the 1 up top does not change the bottom is then the definition of a derivative
Okay, I get that now. How do you change it to f'(x0)?
by definition \[ f'(x_0) = \lim_{x\rightarrow x_0} \frac{y-y_0}{x-x_0} \]
ok, thank you
so the idea is that f'(x0)*g'(y0) = 1 if y0 is 0, then x0 = g(0) so we can say \[ f'(g(0)) \ g'(0)=1 \]
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