Final answer:
To find y′ by implicit differentiation, differentiate both sides of the given equation and solve for dy/dx.
Step-by-step explanation:
To find y′ by implicit differentiation, we first differentiate both sides of the given equation with respect to x. Using the chain rule, the derivative of eˣ/ʸ with respect to x is (eˣ/ʸ)((1/ʸ) dy/dx - (eˣ/ʸ²) dy/dx) and the derivative of 5x - 7y with respect to x is 5 - 7(dy/dx). Equating these derivatives, we can solve for dy/dx to find y′.
Using the given equation eˣ/ʸ = 5x - 7y
(eˣ/ʸ)((1/ʸ) dy/dx - (eˣ/ʸ²) dy/dx) = 5 - 7(dy/dx)
Solving for dy/dx, we get
dy/dx = [(5 - (eˣ/ʸ)((1/ʸ) dy/dx)] / [7 + (eˣ/ʸ²) dy/dx]