Value of $frac{partial }{partial x}left(fleft(x,yright)right)$ at $(0,1)$

Question

$fleft(x,yright)=e^{x+y}left(x^{frac{1}{3}}left(y-1right)^{frac{1}{3}}+y^{frac{1}{3}}left(x-1right)^{frac{2}{3}}right)$

What's the value of $frac{partial }{partial x}left(fleft(x,yright)right)$ at $(0,1)$? There two solutions below. Why are the answers different?
Answer 1:

$fleft(x,1right)=e^{x+1}left(x-1right)^{frac{2}{3}}$

$frac{partial }{partial :x}left(fleft(x,1right)right)=e^{x+1}left(x-1right)^{frac{2}{3}}+frac{2}{3}e^{x+1}left(x-1right)^{-frac{1}{3}}$

so $frac{partial}{partial x}left(fleft(0,1right)right)=frac{e}{3}$

Answer 2：

$lim_{Delta xto 0}frac{fleft(0+Delta x,1right)-fleft(0,1right)}{Delta x}$

$=lim_{Delta xto 0}frac{e^{Delta x+1}left(Delta x-1right)^{frac{2}{3}}-e}{Delta x}$

$lim_{Delta xto 0}frac{eleft(left(Delta x-1right)^{frac{2}{3}}-1right)}{Delta x}$

$=lim_{Delta xto 0}frac{eleft(left(1-Delta xright)^{frac{2}{3}}-1right)}{Delta x}=lim_{Delta xto 0}frac{eleft(1-frac{2}{3}Delta x-1right)}{Delta x}$

$=-frac{2}{3}e$

user247327 · Answer

The first one is wrong!
You correctly have that $f(x,1)= e^{x+1}(x- 1)^{2/3}$
but the derivative of that is $f_x(x,1)= e^{x+1}(x+1)^{2/3}+ (2/3)e^{x+ 1}(x- 1)^{-1/3}$.
You incorrectly has "-" rather than "+".

user10354138 · Answer

In answer 1 you mistakenly go from
$$require{color}
f(x,1)=e^{x+1}(x-1)^{{color{red}2}/3}
$$
to
$$
frac{mathrm{d}}{mathrm{d}x}[f(x,1)]=frac{mathrm{d}}{mathrm{d}x}[e^{x+1}(x-1)^{{color{red}1}/3}]
$$
(plus you forgot the exponential factor).
Doing it correctly would give answer 1 as
begin{align*}
frac{mathrm{d}}{mathrm{d}x}[f(x,1)]&=frac{mathrm{d}}{mathrm{d}x}[e^{x+1}(x-1)^{2/3}]\
&=e^{x+1}(x-1)^{2/3}+frac23e^{x+1}(x-1)^{-1/3}\
thereforeleft.frac{mathrm{d}}{mathrm{d}x}rightvert_{x=0}[f(x,1)]&=e^{0+1}(0-1)^{2/3}+frac23e^{0+1}(0-1)^{-1/3}=dots
end{align*}
Similarly, in answer 2, you forget the factor $e^{Delta x}$ when you go from
$$
lim_{Delta xto 0}frac{e^{Delta x+1}(Delta x-1)^{frac{2}{3}}-e}{Delta x}$$
to
$$lim_{Delta xto 0}frac{ebig((Delta x-1)^{frac{2}{3}}-1big)}{Delta x}$$
when it is, in fact,
$$lim_{Delta xto 0}frac{ebig(e^{Delta x}(Delta x-1)^{frac{2}{3}}-1big)}{Delta x}.$$

Value of $frac{partial }{partial x}left(fleft(x,yright)right)$ at $(0,1)$

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