y = ln((x^2-1)/(x^2+1)) + sin^2(x)/2cos3x
Найдем сначала производную первого слагаемого:
(ln((x^2-1)/(x^2+1)))' = 1/((x^2-1)/(x^2+1))*((x^2-1)/(x^2+1))'=(x^2+1)/(x^2-1) * ((x^2-1)/(x^2+1))' = (x^2+1)/(x^2-1) * ((x^2-1)'(x^2+1) - (x^2-1)(x^2+1)')/(x^2+1)^2 = (x^2+1)/(x^2-1) * (2x(x^2+1) -2x(x^2-1))/(x^2+1)^2= (x^2+1)/(x^2-1) * (2x^3+2x-2x^3+2x)/(x^2+1)^2 = (x^2+1)/(x^2-1) * (4x)/(x^2+1)^2 = 4x/((x^2-1)(x^2+1))= 4x/(x^4-1)
Найдем теперь производную второго слагаемого
= ((sin^2(x))'*2cos3x - sin^2(x)*(2cos3x)')/4cos^2(3x) = (sin2x*2cos3x + sin^2(x) * 2sin3x*3)/4cos^2(3x) = (2sin2xcos3x +6sin3xsin^2(x))/4cos^2(3x) = (sin2xcos3x + 3sin3xsin^2(x))/2cos^2(3x)= (2sinxcosx*cos3x + 3sin3xsin^2(x))/2cos^2(3x) = 1/2* sinx/cos^2(3x) * (2cosxcos3x + 3sinxsin3x)
y'= 4x/(x^4-1) + 1/2 * sinx/cos^2(3x) * (2cosxcos3x + 3sinxsin3x)