if we have functions f and g, and we let h:=sqrt((f^2) + (e^-fg)) calculate ∇h in terms of f,g,∇f, ∇g?
The symbol you use is $\displaystyle \nabla$ which means the gradient vector. Are we to assume these are vector functions? Is so, what is meant by the exponential or the product of two vectors?
If you simply meant "the derivative" and these are all real valued functions of a single variable, then use the chain rule.
$\displaystyle \frac{\partial h}{\partial f}= \frac{1}{2}\left(f^2+ e^{-fg}\right)^{-1/2}(2f- ge^{-fg})$,
$\displaystyle \frac{\partial h}{\partial g}= \frac{1}{2}\left(f^2+ e^{-fg}\right)^{-1/2}(-fe^{-fg})$
and then
$\displaystyle \frac{d h}{d x}= \frac{\partial h}{\partial f}\frac{df}{dx}+ \frac{\partial h}{\partial g}\frac{dg}{dx}$