Answer
$\nabla f = (2xye^{\frac{y}{z}})i + ((x^2e^\frac{y}{z})+ (\frac{x^2ye^\frac{y}{z}}{z}))j + (-\frac{x^2y^2e^\frac{y}{z}}{z^2})k$
Work Step by Step
Derive $f(x,y,z)$ with respect to $x$ to get:
$f_{x}(x, y, z) = 2xye^{\frac{y}{z}}$
Derive $f(x,y,z)$ with respect to $y$ to get:
$f_{y}(x, y, z) = (x^2e^\frac{y}{z})+ \frac{x^2ye^\frac{y}{z}}{z}$
Derive $f(x,y,z)$ with respect to $z$ to get:
$f_{z}(x, y, z) = -\frac{x^2y^2e^\frac{y}{z}}{z^2}$
the gradient of f is calculated as follows: $\nabla f = f_{x}(x, y, z)i + f_{y}(x, y, z) j + f_{z}(x, y, z)k$ $ = (2xye^{\frac{y}{z}})i + ((x^2e^\frac{y}{z})+ \frac{x^2ye^\frac{y}{z}}{z})j + (-\frac{x^2y^2e^\frac{y}{z}}{z^2})k $
