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})?