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if-df-dx-f-what-are-the-value-of-d-dx-Q3707-




Question Number 3764 by 123456 last updated on 19/Dec/15
if (df/dx)=f, what are the value of  (dθ/dx)? (Q3707)
$$\mathrm{if}\:\frac{{df}}{{dx}}={f},\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{{d}\theta}{{dx}}?\:\left(\mathrm{Q3707}\right) \\ $$
Commented by Filup last updated on 19/Dec/15
Attempt:    (dθ/dx)=((d^2 f/dx^2 )/(1+((df/dx))^2 ))  ((df/dx))^2 =f^2   (d^2 f/dx^2 )=f′    ∴(dθ/dx)=((f′)/(1+f^2 ))
$$\mathrm{Attempt}: \\ $$$$ \\ $$$$\frac{{d}\theta}{{dx}}=\frac{\frac{{d}^{\mathrm{2}} {f}}{{dx}^{\mathrm{2}} }}{\mathrm{1}+\left(\frac{{df}}{{dx}}\right)^{\mathrm{2}} } \\ $$$$\left(\frac{{df}}{{dx}}\right)^{\mathrm{2}} ={f}^{\mathrm{2}} \\ $$$$\frac{{d}^{\mathrm{2}} {f}}{{dx}^{\mathrm{2}} }={f}' \\ $$$$ \\ $$$$\therefore\frac{{d}\theta}{{dx}}=\frac{{f}'}{\mathrm{1}+{f}^{\mathrm{2}} } \\ $$
Commented by prakash jain last updated on 20/Dec/15
(df/dx)=f  f(x)=e^(x+c)   tan θ=f ′(x)=e^(x+c)   (dθ/dx)=(e^(x+c) /(1+e^(2x+2c) ))
$$\frac{{df}}{{dx}}={f} \\ $$$${f}\left({x}\right)={e}^{{x}+{c}} \\ $$$$\mathrm{tan}\:\theta={f}\:'\left({x}\right)={e}^{{x}+{c}} \\ $$$$\frac{\mathrm{d}\theta}{\mathrm{d}{x}}=\frac{{e}^{{x}+{c}} }{\mathrm{1}+{e}^{\mathrm{2}{x}+\mathrm{2}{c}} } \\ $$

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