Let $ f: R^{n} \rightarrow R^{p}$ and $ g:R^{n} \rightarrow R^{q}$ be differentiable functions. Define $ B(f,g)(a)=(f(a),g(a)) $, show that $ B(f,g) $ is also differentiable and $DB(f,g)a .(h) =B(Df(a)(h),g(a))+B(f(a),Dg(a)(h)) $. $$ $$ To answer this i have tried a little , If f,g : $U \rightarrow R^{n}$ , $U \subset R^{m}$ , be differentiable and $(f,g)(x)=(f(x),g(x))$, then $(f,g)(x)$ is also differentiable and $ D(f,g)h=(Df (h), Dg(h))$ But in this case please help me,
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What does $B(f(a), Dg(a)(h))$ mean? $f(a)$ and $Dg(a)h$ are both vectors but $B$ is a operator on functions as you defined. – Brian Feb 21 '17 at 03:41
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here B is binlear map – Arifa zaman Feb 21 '17 at 03:42
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1So you exactly say $B(f,g)(a) = B(f(a), g(a))$. Think about $B(f,g)(a+t) - B(f,g)(a) = B(f(a+t), g(a+t)) - B(f(a), g(a+t)) + B(f(a),g(a+t)) - B(f(a),g(a))$ – Brian Feb 21 '17 at 03:50
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then how do i use the information that f and g are differentiable – Arifa zaman Feb 21 '17 at 06:37