Part 2 of 3 Given the equation $$ 9 \sin 5 x y=1 $$, evaluate $$ \frac{d y}{d x} $$. Assume that the equation implicitly defines $$ y $$ as a differentiable function If $$ F(x, y)=9 \sin 5 x y-1 $$, then $$ F_{x}=45 y \cos (5 x y) $$. If $$ F(x, y)=9 \sin 5 x y-1 $$, then $$ F_{y}=\square $$

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UpStudy AI · Accepted Answer

We are given  
$$
F(x, y) = 9 \sin (5xy) - 1.
$$  
To compute $$\frac{\partial F}{\partial y}$$, differentiate with respect to $$y$$ (treating $$x$$ as a constant).

1. Differentiate the outer function:
   $$
   \frac{\partial}{\partial y}[9 \sin (5xy)] = 9 \cos (5xy) \cdot \frac{\partial}{\partial y}(5xy).
   $$
2. Differentiate the inner function:
   $$
   \frac{\partial}{\partial y}(5xy) = 5x.
   $$
3. Combine the results:
   $$
   F_y = 9 \cos (5xy) \cdot 5x = 45x \cos (5xy).
   $$

Thus,  
$$
F_y = 45x \cos (5xy).
$$
$$ F_y = 45x \cos (5xy) $$.

Part 2 of 3 Given the equation \( 9 \sin 5 x y=1 \), evaluate \( \frac{d y}{d x} \). Assume that the equation implicitly defines \( y \) as a differentiable function If \( F(x, y)=9 \sin 5 x y-1 \), then \( F_{x}=45 y \cos (5 x y) \). If \( F(x, y)=9 \sin 5 x y-1 \), then \( F_{y}=\square \)

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