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Question Number 123706 by I want to learn more last updated on 27/Nov/20

If    h(x)  =  f^2 (x)  −  g^2 (x),        f ′(x)   =   − g(x)    and     g ′(x)  =  f(x)     then    h ′(x)  =   ???    (a)    0         (b)     1              (c)     −  4 f(x) g(x)          (d)     [−g(x)]^2   −  [f(x)]^2

$$\mathrm{If}\:\:\:\:\mathrm{h}\left(\mathrm{x}\right)\:\:=\:\:\mathrm{f}^{\mathrm{2}} \left(\mathrm{x}\right)\:\:−\:\:\mathrm{g}^{\mathrm{2}} \left(\mathrm{x}\right),\:\:\:\:\:\:\:\:\mathrm{f}\:'\left(\mathrm{x}\right)\:\:\:=\:\:\:−\:\mathrm{g}\left(\mathrm{x}\right)\:\:\:\:\mathrm{and}\:\:\:\:\:\mathrm{g}\:'\left(\mathrm{x}\right)\:\:=\:\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\:\:\:\mathrm{then}\:\:\:\:\mathrm{h}\:'\left(\mathrm{x}\right)\:\:=\:\:\:??? \\ $$$$ \\ $$$$\left(\mathrm{a}\right)\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\:\:\:\:−\:\:\mathrm{4}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{g}\left(\mathrm{x}\right)\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\:\:\:\:\left[−\mathrm{g}\left(\mathrm{x}\right)\right]^{\mathrm{2}} \:\:−\:\:\left[\mathrm{f}\left(\mathrm{x}\right)\right]^{\mathrm{2}} \\ $$

Answered by Bird last updated on 27/Nov/20

h^′ (x)=2f(x)f^′ (x)−2g(x)g^′ (x)

$${h}^{'} \left({x}\right)=\mathrm{2}{f}\left({x}\right){f}^{'} \left({x}\right)−\mathrm{2}{g}\left({x}\right){g}^{'} \left({x}\right) \\ $$

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