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October 15th, 2018, 06:49 AM   #1
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From: algeria

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Directional derivative of exponential of a function

I have a trouble of a directional derivative of a function :

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})?
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