Let $f ( x , y ) = x ^ { 3 } - 4 \log _ { 7 } \left( x ^ { 2 } \right) + \sin ^ { - 1 } ( x y )$ . Then $f _ { x } ( x , y )$ is

A) $- 3 x ^ { 2 } - \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
B) $3 x ^ { 2 } + \frac { 8 } { x \ln 7 } - \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
C) $3 x ^ { 2 } - \frac { 8 } { x \ln 7 } - \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
D) $3 x ^ { 2 } - \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
E) $3 x ^ { 2 } + \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$

Question

Let $f ( x , y ) = x ^ { 3 } - 4 \log _ { 7 } \left( x ^ { 2 } \right) + \sin ^ { - 1 } ( x y )$ . Then $f _ { x } ( x , y )$ is

A) $- 3 x ^ { 2 } - \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
B) $3 x ^ { 2 } + \frac { 8 } { x \ln 7 } - \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
C) $3 x ^ { 2 } - \frac { 8 } { x \ln 7 } - \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
D) $3 x ^ { 2 } - \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$
E) $3 x ^ { 2 } + \frac { 8 } { x \ln 7 } + \frac { y } { \sqrt { 1 - x ^ { 2 } y ^ { 2 } } }$

Quizplus · Accepted Answer

To find $f_x(x, y)$, we differentiate $f(x, y)$ with respect to $x$, treating $y$ as a constant. The derivative of $x^3$ with respect to $x$ is $3x^2$. For $-4\log_7(x^2)$, using the chain rule and the fact that $\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$, the derivative is $-4 \cdot \frac{2}{x\ln(7)} = -\frac{8}{x\ln(7)}$. For $\sin^{-1}(xy)$, using the chain rule and the derivative of $\sin^{-1}(x)$, which is $\frac{1}{\sqrt{1-x^2}}$, the derivative is $\frac{y}{\sqrt{1-(xy)^2}}$. Combining these gives $3x^2 - \frac{8}{x\ln(7)} + \frac{y}{\sqrt{1-x^2y^2}}$.

Let f(x,y)=x3−4log⁡7(x2)+sin⁡−1(xy)f ( x , y ) = x ^ { 3 } - 4 \log _ { 7 } \left( x ^ { 2 } \right) + \sin ^ { - 1 } ( x y )f(x,y)=x3−4log7​(x2)+sin−1(xy)

Let $f ( x , y ) = x ^ { 3 } - 4 \log _ { 7 } \left( x ^ { 2 } \right) + \sin ^ { - 1 } ( x y )$