derivative100 [31-40]

31. $\frac{d^2}{dx^2}\frac{1}{9} sec(3x)$

$\begin{aligned} &\frac{d^2}{dx^2}\frac{1}{9} sec(3x)\\ &=\frac{1}{9}\frac{d^2}{dx^2} sec(3x)=\frac{1}{3}\frac{d}{dx}sec(3x)tan(3x)\\ &=sec(3x)tan(3x)tan(3x)+sec(3x)sec^2(3x)\\ &=sec(3x)(tan^2(3x)+sec^2(3x))\\ \end{aligned}$

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32. $\frac{d^2}{dx^2} (x+1)/sqrt(x)$

$\begin{aligned} &\frac{d^2}{dx^2} \frac{x+1}{\sqrt x} \\ &=\frac{d}{dx}\frac{\sqrt x-(x+1)\frac{1}{2\sqrt x} }{x}=\frac{d}{dx}\frac{\frac{\sqrt x}{2}-\frac{\sqrt x}{2x}}{x}\\ &=\frac{d}{dx}\frac{x\sqrt x -\sqrt x }{2x^2}= \frac{1}{2}\frac{d}{dx} x^{-\frac{1}{2}}-x^{-\frac{3}{2}}\\ &=-\frac{1}{4}x^{-\frac{3}{2}}+\frac{3}{4}x^{-\frac{5}{2}}\\ &=\frac{3-x}{4x^{\frac{5}{2}}} \end{aligned}$

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33. $\frac{d^2}{dx^2}arcsin(x^2)$

$\begin{aligned} &\frac{d^2}{dx^2}arcsin(x^2) \\ &=\frac{d}{dx} \frac{2x}{\sqrt{1-x^4}}=\frac{2\sqrt{1-x^4}-2x\frac{-4x^3}{2\sqrt{1-x^4}}}{1-x^4}\\ &=\frac{2\frac{1-x^4}{\sqrt{1-x^4}}+\frac{4x^4}{\sqrt{1-x^4}}}{1-x^4}\\ &=\frac{2+2x^4}{(1-x^4)^{3/2}} \end{aligned}$

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34. $\frac{d^2}{dx^2}1/(1+cosx)$

$\begin{aligned} &\frac{d^2}{dx^2} \frac{1}{1+cosx} \\ &=\frac{d}{dx} \frac{sin x}{(1+cos x)^2}=\frac{cos x(1+cosx)^2+sinx2(1+cosx)sinx}{(1+cos x)^4}\\ &=A\\ &=\frac{cosx+2cos^2x+cos^3x+2sin^2x+2sin^2xcosx}{(1+cosx)^4}\\ &=\frac{2+cosx(1+cos^2x+2sin^2x)}{(1+cosx)^4}\\ &=\frac{2+cosx(2+sin^2x)}{(1+cosx)^4}\\ &=\frac{2+cosx(3-cos^2x)}{(1+cosx)^4}=\frac{2+3cosx-cos^3x}{(1+cosx)^4}\\ &A=\frac{cosx(1+cosx)+2sin^2x}{(1+cosx)^3}=\frac{cosx+sin^2x+1}{(1+cosx)^3} \end{aligned}$

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35. $\frac{d^2}{dx^2}(x)arctan(x)$

$\begin{aligned} &\frac{d^2}{dx^2}(x)arctan(x) \\ &=\frac{d}{dx}arctan(x)+x\frac{1}{1+x^2}\\ &=\frac{1}{1+x^2}+\frac{1}{1+x^2}-x(1+x^2)^{-2}2x\\ &=\frac{2}{1+x^2}-\frac{2x^2}{(1+x^2)^2}\\ &=\frac{2+2x^2-2x^2}{(1+x^2)^2}=\frac{2}{(1+x^2)^2}\\ \end{aligned}$

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36. $\frac{d^2}{dx^2}x^4 lnx$

$\begin{aligned} &\frac{d^2}{dx^2}x^4 lnx \\ &=\frac{d}{dx}4x^3lnx+x^3\\ &=4(3x^2)lnx+4x^3(1/x)+3x^2\\ &=12x^2lnx+7x^2 \end{aligned}$

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37. $\frac{d^2}{dx^2}e^{-x^2}$

$\begin{aligned} &\frac{d^2}{dx^2}e^{-x^2} \\ &=\frac{d}{dx} -2xe^{-x^2}=-2e^{-x^2}-2x(-2xe^{-x^2})\\ &=e^{-x^2}(4x^2-2) \end{aligned}$

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38. $\frac{d^2}{dx^2}cos(lnx)$

$\begin{aligned} &\frac{d^2}{dx^2} cos(lnx) =\frac{d}{dx}-sin(lnx)\frac{1}{x}\\ &=-cos(lnx)\frac{1}{x}\frac{1}{x}-sin(lnx)(-1)\frac{1}{x^2}\\ &=\frac{1}{x^2}(sin(lnx)-cos(lnx)) \end{aligned}$

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39. $\frac{d^2}{dx^2}ln(cosx)$

$\begin{aligned} &\frac{d^2}{dx^2} ln(cosx)=\frac{d}{dx}\frac{-sinx}{cosx}\\ &=\frac{d}{dx}-tanx=-sec^2x \end{aligned}$

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40. $\frac{d}{dx} sqrt(1-x^2) + (x)(arcsinx)$

$\begin{aligned} &\frac{d}{dx} \sqrt{1-x^2} + (x)(arcsinx) \\ &=\frac{-2x}{2\sqrt{1-x^2}}+arcsinx+x\frac{1}{\sqrt{1-x^2}}\\ &=arcsin(x) \end{aligned}$

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