The problem appears to be assuming that $Df(x)$ is a linear map from a vector space V with an inner product <,> to $\mathbb{R}$. Correct, yes? If so, then think about the connection between linear maps $V \rightarrow \mathbb{R}$ and the inner product. It helps to look at some more concrete cases for these problems. For example, where V is just Cartesian space $\mathbb{R}^n$. If $L:V \rightarrow \mathbb{R}$ is linear, what would a formula for $L(x_1,x_2, x_3, ...)$ look like? Well, $$L(x_1,x_2,x_3) = 2x_1+x_2-3x_3$$.
Can you express the right-hand side as an inner product of $(x_1,x_2,x_3)$ and some other vector $c$? What would $c$ be if $L(x_1,x_2,x_3)$ is to be $<c,x>$?
Now think about the original question. $Df(x)$ simply denotes some linear map like $L$. Must there always be formula for $L$ like the one in the example?
Start with the above considerations and continue until done ;-)