The derivative of a map $F$ between manifolds $M$ and $N$ is defined by $F_*X(f)= X(f \circ F)$ where $X \in T_P(M)$, the tanget space at the point $P$.
We know that $\left\{\frac{\partial}{\partial x^i}\bigg|_P\right\}_i$ is a basis for $T_P(M)$. How to show that $\left\{dx^i\bigg|_P\right\}_i$ is a basis for the cotangent space $(T_P(M))^*$?
First, by $dx^i$, I guess we mean the derivative of the map $x^i$ as defined above, right? Is this map $x^i$ just picking out the ith coordinate? Secondly, to show that it is a basis, we need to show that $dx^i\left(\frac{\partial}{\partial x^j}\bigg|_P\right) = \delta^i_j.$ Where to go from here: $\underbrace{(dx^i)_P}_{(\Phi_*)_P}\underbrace{\left(\frac{\partial}{\partial x^j}\bigg|_P\right)}_{X}f = \left(\frac{\partial}{\partial x^j}\bigg|_P\right)(f\circ x^i)?$
I can use the chain rule but I am not sure exactly. Please help.