Question

Помогите найти производные

Answer 1

$1); ; y=lnsqrt{ frac{1+sinx}{1-cosx} }\\y'= sqrt{ frac{1-cosx}{1+sinx} } cdot frac{1}{2sqrt{frac{1+sinx}{1-cosx}}} cdot frac{cosx(1-cosx)-(1+sinx)sinx}{(1-cosx)^2} =\\= frac{1-cosx}{2(1+sinx)} cdot frac{cosx-cos^2x-sinx-sin^2x}{(1-cosx)^2} = frac{cosx-sinx-1}{2(1+sinx)(1-cosx)}\\2); ; y=log_a(x^2+1)\\y'= frac{2x}{(x^2+1)lna}$

$3); ; y=ln(lnx)\\y'= frac{1}{lnx} cdot frac{1}{x} \\4); ; y=ln frac{sqrt{x^2+1}-x}{sqrt{x^2+1}+x} \\y'= frac{sqrt{x^2+1}+x}{sqrt{x^2+1}-x}cdot frac{(frac{2x}{2sqrt{x^2+1}}-1)(sqrt{x^2+1}+x)-(sqrt{x^2+1}-x)(frac{2x}{2sqrt{x^2+1}}+1)}{(sqrt{x^2+1}+x)^2} \\5); ; y=7^{x^2+2x}\\y'=7^{x^2+2x}cdot ln7cdot (2x+2)\\6); ; y=a^{tg, nx}\\y'=a^{tg, nx}cdot lnacdot frac{n}{cos^2nx}$

$7); ; y=(sinx)^{x}\\lny=xcdot ln(sinx)\\ frac{y'}{y}=ln(sinx)+xcdot frac{cosx}{sinx} =ln(sinx)+xcdot ctgx\\y'=(sinx)^{x}cdot (ln(sinx)+xcdot ctgx)\\8); ; y= frac{1-e^{x}}{1+e^{x}} \\y'= frac{-e^{x}(1+e^{x})-(1-e^{x})e^{x}}{(1+e^{x})^2} = frac{-e^{x}-e^{2x}-e^{x}+e^{2x}}{(1+e^{x})^2} =-frac{2e^{x}}{(1+e^{x})^2}$