Hello

I have a question if it possible

for a function f and a tangent vector field X of a manifold M, we have

X(f)=∇_{X}f=<∇f,X>

From this above formula, is it true that :

X(e^{-f})=<∇e^{-f},X>=<e^{-f}∇(-f),X>=-e^{-f}<∇f,X>

If it not true, what is X(e^{-f})?