Partial derivatives

Question

Find the first partial derivatives of the function.

g(x, y) = 5 ln(4x + ln y) 



g_x(x, y) =  

g_y(x, y) =  

Answer 1

To find the partial derivatives of the function g(x, y) with respect to x and y, we will differentiate g(x, y) with respect to x and y, respectively, while treating y as a constant when differentiating with respect to x, and x as a constant when differentiating with respect to y.

So, we have:

∂g/∂x = ∂/∂x [5 ln(4x + ln y)] = 5 * d/dx [ln(4x + ln y)] // using the chain rule = 5 * 1/(4x + ln y) * d/dx [4x + ln y] // using the chain rule = 5 * 1/(4x + ln y) * 4 = 20/(4x + ln y)

And,

∂g/∂y = ∂/∂y [5 ln(4x + ln y)] = 5 * d/dy [ln(4x + ln y)] // using the chain rule = 5 * 1/(4x + ln y) * d/dy [4x + ln y] // using the chain rule = 5 * 1/(4x + ln y) * 1/y = 5/(y(4x + ln y))

Therefore, the first partial derivatives of the function g(x, y) are:

∂g/∂x = 20/(4x + ln y) ∂g/∂y = 5/(y(4x + ln y))