Question

Y=arccos√1-e^2x. Найти производные

Answer 1

$y=arccos(sqrt{1-e^{2x}})\\y=f(g(h(u(x)))):\f(t)=arccos(t)\g(s)=sqrt{s}\h(q)=1-e^q\u(x)=2x\\y'=f'(g(h(u(x))))*g'(h(u(x)))*h'(u(x))*u'(x)\\f'(t)=-frac{1}{sqrt{1-t^2}}\g'(s)=frac{1}{2sqrt{s}}\h'(q)=-e^q\u'(x)=2$

$y'=(-frac{1}{sqrt{1-(sqrt{1-e^{2x}})^2}})(frac{1}{2sqrt{1-e^{2x}}})(-e^{2x})(2)=\\frac{2e^{2x}}{2sqrt{1-e^{2x}}}(frac{1}{sqrt{1-(1-e^{2x})}})=frac{e^{2x}}{sqrt{1-e^{2x}}(sqrt{(e^x)^2})}=frac{(e^{x})^2}{e^xsqrt{1-e^{2x}}}=frac{e^x}{sqrt{1-e^{2x}}}$