derivative100 [61-70]

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

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

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62. $\frac{d}{dx} \frac{sinx-cosx}{sinx+cosx}$

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

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63. $\frac{d}{dx}4x^2(2x^3 – 5x^2)$

$\begin{aligned} &\frac{d}{dx}4x^2(2x^3 – 5x^2)\\ &=8x(2x^3-5x^2)+4x^2(6x^2-10x)\\ &=40x^4-80x^3=40x^3(x-2) \end{aligned}$

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64. $\frac{d}{dx}(\sqrt x)(4-x^2)$

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

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65. $\frac{d}{dx} \sqrt{\frac{1+x}{1-x}}$

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

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66. $\frac{d}{dx}sin(sinx)$

$\begin{aligned} &\frac{d}{dx}sin(sinx)\\ &=cos(sinx)cosx \end{aligned}$

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67. $\frac{d}{dx}(1+e^{2x})/(1-e^{2x})$

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

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68. $\frac{d}{dx}[x/(1+lnx)]$

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

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69. $\frac{d}{dx}x^{x/lnx}$

$\begin{aligned} &\frac{d}{dx}x^{x/lnx}\\ &y=x^{x/lnx}, lny=\frac{x}{lnx}lnx=x,\frac{1}{y}dy=dx\\ &\frac{dy}{dx}=y=x^{x/lnx}\\ &=e^{{lnx}^{x/lnx}}=e^{(x/lnx)lnx}=e^x \end{aligned}$

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70. $\frac{d}{dx}ln[\sqrt{\frac{x^2-1}{x^2+1}}]$

$\begin{aligned} &\frac{d}{dx}ln[\sqrt{\frac{x^2-1}{x^2+1}}]\\ &=\sqrt \frac{x^2+1}{x^2-1}\frac{\frac{2x}{2\sqrt{x^2-1}}\sqrt{x^2+1}-\sqrt{x^2-1}{\frac{2x}{2\sqrt{x^2+1}}}}{x^2+1}\\ &=\frac{ \frac{x(x^2+1)}{\sqrt{x^2-1}} -x\sqrt{x^2-1} }{\sqrt{x^2-1}(x^2+1)}\\ &=\frac{x^3+x-x^3+x}{(x^2-1)(x^2+1)}=\frac{2x}{(x^2-1)(x^2+1)} \end{aligned}$
Alt.
$ln[\sqrt{\frac{x^2-1}{x^2+1}}]=\frac{1}{2}ln(x^2-1)-\frac{1}{2}ln(x^2+1)$

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