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 of $$ x $$. If $$ F(x, y)=9 \sin 5 x y-1 $$, then $$ F_{x}=\square $$

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UpStudy AI · Accepted Answer

We have

$$
F(x,y)=9 \sin (5xy)-1.
$$

To compute the partial derivative with respect to $$x$$, we differentiate $$F(x,y)$$ treating $$y$$ as a constant:

1. Differentiate $$9 \sin (5xy)$$ with respect to $$x$$:
   $$
   \frac{\partial}{\partial x}\left(9 \sin (5xy)\right)=9 \cos (5xy) \cdot \frac{\partial}{\partial x}(5xy).
   $$

2. Since $$y$$ is constant, the derivative of $$5xy$$ with respect to $$x$$ is:
   $$
   \frac{\partial}{\partial x}(5xy)=5y.
   $$

3. Multiplying, we obtain:
   $$
   \frac{\partial}{\partial x}\left(9 \sin (5xy)\right)=9 \cdot 5y \cdot \cos (5xy)=45y\cos (5xy).
   $$

4. The derivative of the constant $$-1$$ is $$0$$.

Thus, the final result is

$$
F_{x}(x,y)=45y\cos (5xy).
$$
$$ F_{x}(x,y) = 45y \cos(5xy) $$

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 of \( x \). If \( F(x, y)=9 \sin 5 x y-1 \), then \( F_{x}=\square \)

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