I have been asked to describe the tangents space $T_q(df(M))$ as a subspace of $T_q(T^*M)$ where $f\in C^\infty(M)$ and $df$ is a 1-form (or smooth section of $T^*M$).
Here, $df:M\rightarrow T^*M$ is a function and $df(M)$ is an image of $M$ under $df$. And this image is naturally a smooth manifold because $df$ is smooth embedding.
But I have no idea on it. What should I focus on to describe it? Even it is difficult to see why $df(M)$ is a smooth manifold. I will be very appreciate for any help. Thanks in advance.
Newly Added
I started to think how $T_q(T^*M)$ look like. Suppose that the dimension of $M$ is $n$.
Note that $\exists $ a chart $(U,\varphi=\{x^1,\dots, x^n\})$ such that $q=(p,v)\in T^*U$. Then we have $$v=\sum_{i=1}^{n}y^idx^i. $$
Thus, we can describe local coordinate function of $(p,v)$ as $(x^1,\dots,x^n,y^1,\dots, y^n)$. It implies that our basis of $T_p(T^*M)$ would be $$\left\{\frac{\partial}{\partial x^1},\dots ,\frac{\partial}{\partial x^n} ,\frac{\partial}{\partial y^1},\dots ,\frac{\partial}{\partial y^n} \right\} .$$
I think I can use this to describe $T_q(df(M))$. Currently my concern is that the dimension of $df(M)$ is $n$ since it is diffeomorphic to $M$. In other words, we should pick $n$ element that would be linear combination of the basis above. But I am still working on.
Added(12/5/2018)
Now, consider $T_qL=T_q(df(M))$. Let $q=(p,df(p))\in df(M)$. And note that $df(p)\in T^*M$ which means that \begin{align*} df(p)=\sum_{i=1}^n y^i dx^i \end{align*} where \begin{align*} y^i = df(p)\left( \frac{\partial}{\partial x^i } \right)=\frac{\partial f}{\partial x^i}\bigg|_{p}. \end{align*}
Observe that each $y^i$ is determined by $x^i$. In other words, each $y^i$ is a function of $x^i$. Therefore, $\forall p\in M$, we can represent $(p,df(p))$ as $(x^1,\dots, x^n)$. Thus, our basis for $T_qL$ would be \begin{align*} \left\{ \frac{\partial}{\partial x^1},\dots,\frac{\partial}{\partial x^n}\right\}. \end{align*}
Thus, $T_qL$ is a set of linear combination of the basis above.
But I am uncertain about it. I am trying to making sure if it is right.
Let $(U,\varphi=\{x^i\})$ be the local chart where $p\in U$ and $q=(p,v)\in T^*U$. Then $$ df = \sum_{i=1}^{n}\frac{\partial f}{\partial x^i} dx^i.$$
Then we have the local coordinate $\left\{ x^i,y^i=\frac{\partial f}{\partial x^i} \right\}$ of $T^*M$. Then
$$T_qL=span\left\{ \frac{\partial}{\partial x^i} + \sum_{k=1}^{n}\frac{\partial^2f}{\partial x^i\partial x^k} \frac{\partial}{\partial y^k} \right\}_{i=1}^{n}. $$
In order to understand, think about easy example when $f:\mathbb{R}\rightarrow \mathbb{R}$. If we look at the graph of $f$, for given $p\in \mathbb{R}$, the graph of tangent line at $p$, $$y=f'(p)(x-p)+f(p) $$ is $span\left\{ \overrightarrow{e_1}+\frac{\partial f}{\partial x}\overrightarrow{e_2} \right\}$ where $\{e_1,e_2\}$ is a standard basis of $\mathbb{R}^2$.