Question

17. Consider the function f(x, y, z) = xy ln(yz).

Answer 1

We can expand

$f(x,y,z) = xy\ln(yz) = xy \ln(y) + xy\ln(z)$

Then using the product and chain rules,

$\dfrac{\partial f}{\partial x} = y\ln(y) + y\ln(z) + \boxed{y\ln(yz)}$

$\dfrac{\partial f}{\partial y} = x\ln(y) + \dfrac{xy}y + x\ln(z) = \boxed{x\ln(yz) + x}$

$\dfrac{\partial f}{\partial z} = \boxed{\dfrac{xy}z}$