Maths question involving differentiation.?

Question

Hi

could someone plz help me with that question.

Given the function z=Ln(y/x). find the partial derivatives.

dz/dx and dz/dy.

thank you

Answer 1

Ln(y/x) = Ln(y) - Ln(x)

dz/dx = -1/x

dz/dy = 1/y

Answer 2

In partial differentiation, you treat the variable that is not being differentiated as a constant.

Just a note, the derivative of ln(x) is 1/x.

Knowing that, let us proceed.

dz/dx = 1 all over y/x multiplied by the derivative of y/x

dz/dx = x/y multiplied by -y/x^2 (derivative of 1/x is -1/x^2 and treating y as a constant, the derivative f y/x is -y/x^2)

y and x can be canceled thereby giving you:

dz/dx = -1/x

Using the same process for dz/dy...

dz/dy = 1 all over y/x multiplyed by the derivative of y/x

dz/dy = x/y multiplied by 1/x (derivative of y is 1. x is treated as a constant thereby giving 1/x)

x can be canceled thereby giving you:

dz/dy = 1/y