Let [tex]L(A;B)[/tex] be the space of linear maps [tex]l:A\rightarrow B[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

My goal is to derive the Leibniz (Product) Rule using the chain rule. Let [tex]f_i:U\subset E\rightarrow F_i, i=1,2[/tex] be differentiable maps and let [tex]B\in L(F_1,F_2;G)[/tex]. Then the mapping [tex]B(f_1,f_2)=B\circ (f_1\times f_2):U\subset E\rightarrow G[/tex] is differentiable.

Using the chain rule I find for [tex]u\in U[/tex]: [tex]D \{B\circ (f_1\times f_2)\}(u)=DB \{(f_1\times f_2)(u)\}\circ D(f_1\times f_2)(u)=B\circ D(f_1\times f_2)(u)\in L(E;G)[/tex], as B is a linear map. I could now go on and take the derivative of the cartesian product: [tex]...=B\circ(Df_1\times Df_2)(u)[/tex], but I feel it's all wrong. To me, there's no product rule in sight. Could you point out my error(s)?

Thanks alot. Best regards...Cliowa

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# Differentiation on banach spaces

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