Derivative100 [81-90]

81. $\frac{d}{dx}e^x sinhx$

$\begin{aligned} &\frac{d}{dx}e^x sinhx\\ &=e^xsinhx+e^xcoshx=e^x(sinhx+coshx)\\ &=e^x(\frac{2e^x}{2})=e^{2x} \end{aligned}$

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82. $\frac{d}{dx}sech(\frac{1}{x})$

$\begin{aligned} &\frac{d}{dx}sech(\frac{1}{x})\\ & sech (x)=1/cosh(x), D(sech(x))=-sinh(x)/cosh^2(x)\\ &=-sech(x)tanh(x)\\ &\frac{d}{dx}sech(\frac{1}{x})=-sinh(1/x)/cosh^2(1/x)(-1/x^2)\\ &=\frac{sinh(1/x)}{x^2cosh^2(1/x)}=\frac{sech(\frac{1}{x})tanh(\frac{1}{x})}{x^2}\\ \end{aligned}$

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83. $\frac{d}{dx}cosh(lnx))$

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

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84. $\frac{d}{dx}ln(coshx)$

$\begin{aligned} &\frac{d}{dx}ln(coshx)=\frac{sinhx}{coshx}=tanh(x)\\ \end{aligned}$

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85. $\frac{d}{dx}\frac{sinhx}{1+coshx}$

$\begin{aligned} &\frac{d}{dx}\frac{sinhx}{1+coshx}=\frac{coshx(1+coshx)-sinhxsinhx}{1+cosh^2x+2cosh(x)}\\ &=\frac{coshx+cosh^2x-sinh^2x}{(1+coshx)^2}=\frac{1+coshx}{(1+coshx)^2}\\ &=\frac{1}{1+coshx} \end{aligned}$

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86. $\frac{d}{dx}arctanh(cosx)$

$\begin{aligned} &\frac{d}{dx}arctanh(cosx)\\ &D(arctanh(x)) = \frac{1}{1-x^2}\\ &y=arctanh(x), x=tanh(y), dx=sech^2(y)dy\\ &dy/dx = 1/sech^2(y)=cosh^2(y)=\frac{1}{(1/cosh^2(y))}\\ &=\frac{1}{( cosh^2(y)-sinh^2(y))/cosh^2(y)}=\frac{1}{1-tanh^2(y)}\\ &=\frac{1}{1-x^2}\\ &\frac{d}{dx}arctanh(cosx)=-\frac{sinx}{1-cos^2x}\\ &=-\frac{sin(x)}{sin^2(x)}=-csc(x)\\ \end{aligned}$

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87. $\frac{d}{dx}(x)(arctanhx)+ln(\sqrt{(1-x^2}))$

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

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88. $\frac{d}{dx}arcsinh(tanx)$

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

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89. $\frac{d}{dx}arcsin(tanhx)$

$\begin{aligned} &\frac{d}{dx}arcsin(tanhx)\\ &=\frac{1}{\sqrt{1-tanh^2x}} sech^2x\\ &1-tanh^2x = \frac{cosh^2x-sinh^2x}{cosh^2x}=sech^2x\\ &=\frac{sech^2x}{\sqrt {sech^2x} }=sech(x) \end{aligned}$

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90. $\frac{d}{dx} \frac{arctanhx}{1-x^2}$

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

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