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Question Number 184735 by mustafazaheen last updated on 11/Jan/23
  f(x,y)=((√(3xy^2 )))(((x^5 y^2 ))^(1/5) )  f^′ (x,y)=?     f′′(x,y)=?
$$ \\ $$$${f}\left({x},{y}\right)=\left(\sqrt{\mathrm{3}{xy}^{\mathrm{2}} }\right)\left(\sqrt[{\mathrm{5}}]{{x}^{\mathrm{5}} {y}^{\mathrm{2}} }\right) \\ $$$${f}^{'} \left({x},{y}\right)=?\:\:\:\:\:{f}''\left({x},{y}\right)=? \\ $$$$ \\ $$
Commented by mr W last updated on 11/Jan/23
there are no things like f′(x,y) or  f′′(x,y). if you mean ((∂f(x,y))/∂x) or  ((∂^2 f(x,y))/∂x^2 ), you should also correctly  write what you want to ask.
$${there}\:{are}\:{no}\:{things}\:{like}\:{f}'\left({x},{y}\right)\:{or} \\ $$$${f}''\left({x},{y}\right).\:{if}\:{you}\:{mean}\:\frac{\partial{f}\left({x},{y}\right)}{\partial{x}}\:{or} \\ $$$$\frac{\partial^{\mathrm{2}} {f}\left({x},{y}\right)}{\partial{x}^{\mathrm{2}} },\:{you}\:{should}\:{also}\:{correctly} \\ $$$${write}\:{what}\:{you}\:{want}\:{to}\:{ask}. \\ $$
Commented by mustafazaheen last updated on 11/Jan/23
Mr    ((∂^2 f(x,y))/∂x^2 )=?
$${Mr}\:\:\:\:\frac{\partial^{\mathrm{2}} {f}\left({x},{y}\right)}{\partial{x}^{\mathrm{2}} }=? \\ $$
Commented by mr W last updated on 11/Jan/23
f(x,y)=(√3)x^(3/2) y^(7/5)   (∂f/∂x)=((3(√3))/2)x^(1/2) y^(7/5)   (∂^2 f/∂x^2 )=((3(√3))/4)x^(−(1/2)) y^(7/5)
$${f}\left({x},{y}\right)=\sqrt{\mathrm{3}}{x}^{\frac{\mathrm{3}}{\mathrm{2}}} {y}^{\frac{\mathrm{7}}{\mathrm{5}}} \\ $$$$\frac{\partial{f}}{\partial{x}}=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}}{x}^{\frac{\mathrm{1}}{\mathrm{2}}} {y}^{\frac{\mathrm{7}}{\mathrm{5}}} \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{4}}{x}^{−\frac{\mathrm{1}}{\mathrm{2}}} {y}^{\frac{\mathrm{7}}{\mathrm{5}}} \\ $$

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