Question

If In(xy) = x+y, find y'
А. -y/x
B. e^(x+y)
C.xy/(1-xy)
D. (xy-y)/(x-xy)

Answer 1

To find the derivative y' given ln(xy) = x + y, implicit differentiation is applied. After simplifying, the correct answer is D. (xy - y)/(x - xy).

Step-by-step explanation:

The question asks for the derivative of y with respect to x, given that ln(xy) = x + y. To solve for y', also known as dy/dx, we apply implicit differentiation to both sides of the equation.

Starting with the original equation ln(xy), we apply the product rule of logarithms to separate the variables, which gives us ln(x) + ln(y). Upon differentiating both sides with respect to x, we get:

1. 1/x + (1/y)y' = 1 + y'

We solve for y' by rearranging the terms, and after simplification, we obtain:

1. y' = -y/x / (1 - y/x)
2. y' = -y / (x - xy)
3. Therefore, the correct answer is D. (xy - y)/(x - xy).