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let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If $F(x)=f(x) g(x), G(x)=f^{\prime}(x) g^{\prime}(x)$ and $F^{\prime}(x)=G(x) H(x)=F(x) K(x)$, then $H(x)+K(x)=$
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The correct answer is:
$\frac{f^{\prime}}{f}+\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}}+\frac{g^{\prime}}{g}$
We have, $F(x)=f(x) g(x)$
$\begin{aligned}
& \Rightarrow \quad F^{\prime}(x)=f^{\prime}(x) g(x)+f(x) g^{\prime}(x) \text { and } \\
& G(x)=f^{\prime}(x) g^{\prime}(x) \\
& \text { Now, } F^{\prime}(x)=G(x) H(x) \\
& \therefore \quad H(x)=\frac{F^{\prime}(x)}{G(x)}=\frac{f^{\prime}(x) g(x)+f(x) g^{\prime}(x)}{f^{\prime}(x) g^{\prime}(x)}=\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}} \\
& \text { Again, } F^{\prime}(x)=F(x) K(x) \\
& \Rightarrow \quad K(x)=\frac{F^{\prime}(x)}{F(x)}=\frac{f^{\prime}(x) g(x)+f(x) g^{\prime}(x)}{f(x) g(x)}=\frac{f^{\prime}}{f}+\frac{g^{\prime}}{g} \\
& \therefore H(x)+K(x)=\frac{f^{\prime}}{f}+\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}}+\frac{g^{\prime}}{g}
\end{aligned}$
$\begin{aligned}
& \Rightarrow \quad F^{\prime}(x)=f^{\prime}(x) g(x)+f(x) g^{\prime}(x) \text { and } \\
& G(x)=f^{\prime}(x) g^{\prime}(x) \\
& \text { Now, } F^{\prime}(x)=G(x) H(x) \\
& \therefore \quad H(x)=\frac{F^{\prime}(x)}{G(x)}=\frac{f^{\prime}(x) g(x)+f(x) g^{\prime}(x)}{f^{\prime}(x) g^{\prime}(x)}=\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}} \\
& \text { Again, } F^{\prime}(x)=F(x) K(x) \\
& \Rightarrow \quad K(x)=\frac{F^{\prime}(x)}{F(x)}=\frac{f^{\prime}(x) g(x)+f(x) g^{\prime}(x)}{f(x) g(x)}=\frac{f^{\prime}}{f}+\frac{g^{\prime}}{g} \\
& \therefore H(x)+K(x)=\frac{f^{\prime}}{f}+\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}}+\frac{g^{\prime}}{g}
\end{aligned}$
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