# Thread: Gradient of function of functions

1. ## Gradient of function of functions

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?

2. I can't read your function for $\displaystyle \displaystyle h$.

Is it $\displaystyle \displaystyle h = \sqrt{f^2 + e^{-fg}}$ or $\displaystyle \displaystyle h = \sqrt{f^2 + e^{-f}g}$?

3. Hi, sorry it's $\displaystyle \displaystyle h = \sqrt{f^2 + e^{-fg}}$

4. 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
$\displaystyle \frac{d h}{d x}= \frac{\partial h}{\partial f}\frac{df}{dx}+ \frac{\partial h}{\partial g}\frac{dg}{dx}$