Explanation:
To find the derivative of the given equation with respect to x, we'll use implicit differentiation. Let's go step by step:
Given equation: xe^y - 3y*sin(x) = 1
1. Differentiate both sides with respect to x:
Differentiate the left-hand side of the equation using the product rule and chain rule:
d/dx (xe^y) - d/dx (3y*sin(x)) = d/dx(1)
e^y * (x * dy/dx + 1) - 3y' * sin(x) - 3y * cos(x) = 0
2. Solve for y':
Rearrange the equation to solve for y':
e^y * (x * dy/dx + 1) - 3y' * sin(x) - 3y * cos(x) = 0
e^y * x * dy/dx + e^y - 3y' * sin(x) - 3y * cos(x) = 0
Now isolate the term with y':
e^y * x * dy/dx = 3y' * sin(x) + 3y * cos(x) - e^y
Divide both sides by (e^y * x):
dy/dx = (3y' * sin(x) + 3y * cos(x) - e^y) / (e^y * x)
So, the derivative of y with respect to x, y'(x), is:
y'(x) = (3y' * sin(x) + 3y * cos(x) - e^y) / (e^y * x)