Question

求下列函数导数 dy/dxy=f(sin^2 x)+f(cos^2 x) 其中f(x) 可导 .

Answer 1

y'=[f(sin^2 x)]'+[f(cos^2 x)]'

=f'(sin^2 x)*(sin^2 x)'+f'(cos^2 x)*(cos^2 x)'

=f'(sin^2 x)*2sinx*(sinx)'+f'(cos^2 x)*2cosx*(cosx)'

=f'(sin^2 x)*2sinx*cosx+f'(cos^2 x)*2cosx*(-sinx)

=sin2xf'(sin^2 x)-sin2xf'(cos^2 x)

=sin2x*[f'(sin^2 x)-f'(cos^2 x)]

所以dy/dx=sin2x*[f'(sin^2 x)-f'(cos^2 x)]