Question

The derivative of xy = c² with respect to x is ______.

Options

• `dy/dx = (2c - y)/x`

• `dy/dx = c^2/x^2`

• `dy/dx = (-y)/x`

• `dy/dx = y/x`

Answer 1

The derivative of xy = c² with respect to x is `underlinebb(dy/dx = (-y)/x)`.

Explanation:

The derivative of the equation xy = c² with respect to x is found using implicit differentiation and the product rule.

The product rule states that `d/dx (uv) = u (dv)/(dx) + v (du)/(dx)`.

Applying this to the equation xy = c², where u = x and v = y:

`d/dx (xy) = d/dx (c^2)`

`x xx dy/dx + y xx dx/dx = 0`

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

`x dy/dx + y = 0`

Now, solve for `dy/dx`:

`x dy/dx = -y`

`dy/dx = (-y)/x`