Question

use the equation
\frac{d y}{d x}=-\frac{\frac{partial f}{partial x}}{\frac{partial f}{partial y}}=-\frac{f_{x}}{f_{y}}
to find ( \frac{d y}{d x} ).
cos (x y)=5+sin (y)
( \frac{d y}{d x}= )

Explanation:

Step1: Define \( F(x,y) \)

Let \( F(x,y)=\cos(xy)-\sin(y) - 5 \).

Step2: Calculate \( F_x \)

Using the chain rule, \( F_x =-\sin(xy)\cdot y\).

Step3: Calculate \( F_y \)

Using the chain rule, \( F_y=-\sin(xy)\cdot x-\cos(y)\).

Step4: Apply the formula \( \frac{dy}{dx}=-\frac{F_x}{F_y} \)

Substitute \( F_x \) and \( F_y \) into the formula:

$$
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$$

Answer:

\( \frac{dy}{dx}=-\frac{y\sin(xy)}{x\sin(xy)+\cos(y)} \)