Problem 1279 (difficulty: 4/10)

Let \(\displaystyle H\subset\R^{p+q}\), \(\displaystyle a\in\R^p\), \(\displaystyle b\in\R^q\), \(\displaystyle (a,b)\in\INT H\) and \(\displaystyle f:H\to\R\) differentiable at \(\displaystyle (a,b)\) and assume that near \(\displaystyle a\) there is a differentiable function \(\displaystyle \varphi\) to \(\displaystyle \R^q\) such that \(\displaystyle f(x,\varphi(x))=0\). Prove that

\(\displaystyle f_a'(b) \circ \varphi'(a) = -(f^b)'(a). \)

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