Final answer:
To find dy/dx by implicit differentiation for the equation ex/y = 2x - y, differentiate both sides of the equation. Use the quotient rule on the left side and differentiate the right side. Set the derivatives equal to each other and solve for dy/dx.
Step-by-step explanation:
To find dy/dx by implicit differentiation for the equation ex/y = 2x - y, we can differentiate both sides of the equation with respect to x, treating y as a function of x.
Let's start by differentiating the left side of the equation. Using the quotient rule, we have:
d/dx (ex/y) = (d/dx (ex)y - ex(d/dx (y)))/y^2
Now, let's differentiate the right side of the equation. The derivative of 2x - y with respect to x is simply 2. So:
d/dx (2x - y) = 2
Setting the derivatives of both sides equal to each other, we get:
(d/dx (ex)y - ex(d/dx (y)))/y^2 = 2
Simplifying and isolating d/dx (y), we can solve for dy/dx:
d/dx (y) = (d/dx (ex)y - 2y^2)/ex