Question

If y is a function of x and log (x + y) = xy then the value of `(dy/dx)` at x = 0 is ______.

Options

• 1

• –1

• 2

• 0

Answer 1

If y is a function of x and log (x + y) = xy then the value of `(dy/dx)` at x = 0 is 0.

Explanation:

Given:

log (x + y) = xy

We need `dy/dx` at x = 0

Step 1: First find y when x = 0

Substitute x = 0:

log (0 + y) = 0

log y = 0

y = 1

So the point is (0, 1).

Step 2: Differentiate implicitly

log (x + y) = xy

Differentiate both sides w.r.t. x:

`1/(x + y) (1 + dy/dx) = x dy/dx + y`

Step 3: Substitute x = 0, y = 1

`1/1 (1 + dy/dx) = 0 + 1`

`1 + dy/dx = 1`

`dy/dx = 0`