Derivative of f(x)^(g(x)) at points when f(x)=0 I am interested in understanding the general behavior of the derivative for f(x)^(g(x)) at points where f(x)=0. For example, if f^g=x^n we have (d)/(dx)f^g(0)={(0,n>=1),(+-oo, n<1):} The general formula (f^g)′=f^g(g′ln|f|+g(f′)/(f)) breaks down when f=0, though as the example above shows the derivative may still exist there. I am not sure what the proper assumptions should be. Tentatively, take f(x)^g(x)>=0 for all x in the relevant domain, so that (I believe) we have ln(f(x)^(g(x)))=g(x)ln|f(x)| when f is strictly positive and undefined otherwise.Also, is there any non-trivial example of a well-defined function (meaning "nice", as in differentiable almost everywhere) f^g where f takes on negative values and where g is not constant? In other

$\frac{20b}{{\left(4b^3\right)}^3}$

Which operation could we perform in order to find the number of milliseconds in a year??
${\displaystyle 60\cdot 60\cdot 24\cdot 7\cdot 365}$
${\displaystyle 1000\cdot 60\cdot 60\cdot 24\cdot 365}$
${\displaystyle 24\cdot 60\cdot 100\cdot 7\cdot 52}$
${\displaystyle 1000\cdot 60\cdot 24\cdot 7\cdot 52?}$

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A) No
B) 0
C) Yes
D) 6

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149600000000 is equal to
A)$1.496\times <!--\; \times \; -->{10}^{11}$
B)$1.496\times <!--\; \times \; -->{10}^{10}$
C)$1.496\times <!--\; \times \; -->{10}^{12}$
D)$1.496\times <!--\; \times \; -->{10}^8$

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