Higher order derivative of a composition of vector valued multivariate functions

Question

I have this composition of twice differentiable functions : $$ \Bbb R ^m \overset{f}{\longrightarrow} \Bbb R^n \overset{g}{\longrightarrow} \Bbb R^k$$ And I'm trying to get this result from a book : $$ (g \circ f)''(x) (h,k) = (g''\circ f')(x)\big(f'(x)h,f'(x)k\big) + (g'\circ f)(x)\big(f''(x)(h,k)\big) $$ A poor attempt : $$ \begin{align} (g \circ f)^{\prime\prime}(x) (h,k) & = \bigg(\big((g' \circ f)(x) f'(x)\big)(h)\bigg)'(k) \\ & = \big((g' \circ f)(x)\big)'(k) f'(x)(h) + (g' \circ f)(x)\big(f''(x)(h,k)\big) \\ & = (g'' \circ f)(x)f'(x)(k) f'(x)(h) + (g' \circ f)(x)\big(f''(x)(h,k)\big) \\ & = (g''\circ f \space)(x)\big(f'(x)k,f'(x)h\big) + (g' \circ f)(x)\big(f''(x)(h,k)\big) \\ (?) & = (g''\circ f')(x)\big(f'(x)h,f'(x)k\big) + (g'\circ f)(x)\big(f''(x)(h,k)\big) \end{align}$$ So It feels like it follows the rules of bilinear differentiation but then $h$ and $k$ should swap places and there's that extra $f'$. What am I doing wrong.