Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$ . Then $f _ { x x } ( x , y )$ is

A) $- \frac { 6 y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
B) $\frac { 6 y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
C) $- \frac { 6 x ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
D) $\frac { 6 x ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
E) $- \frac { 6 x y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$

Question

Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$ . Then $f _ { x x } ( x , y )$ is

A) $- \frac { 6 y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
B) $\frac { 6 y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
C) $- \frac { 6 x ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
D) $\frac { 6 x ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$
E) $- \frac { 6 x y ^ { 3 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$

Quizplus · Accepted Answer

To find $f_{xx}(x, y)$, we first differentiate $f(x, y)$ with respect to $x$ to get $f_x(x, y)$, and then differentiate $f_x(x, y)$ with respect to $x$ again to get $f_{xx}(x, y)$. The first derivative $f_x(x, y)$ involves applying the chain rule to the function inside the square root, resulting in a derivative that includes $x$ in the numerator. The second derivative $f_{xx}(x, y)$ will further differentiate this expression with respect to $x$, but none of the provided options correctly represent the process of taking these derivatives step by step for this function. The correct process involves differentiating $6x/(2\sqrt{3x^2 - 2y^3})$ again with respect to $x$, which would simplify to a form not directly provided in the options. However, the misunderstanding in the explanation indicates a need to correct the approach: The correct calculation involves applying the chain rule and the power rule correctly, and none of the options directly match the expected outcome of such a calculation for $f_{xx}$. The mistake in the explanation suggests a reevaluation of the differentiation steps and the application of derivative rules.

Let f(x,y)=3x2−2y3f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }f(x,y)=3x2−2y3​

Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$