Question

Найти f'(x) и f'(x0)
f(x)= xctgx, x0= π/4

Answer 1

((uv)´=u´v+uv´ )

f(x)=xctgx , u=x, v=ctgx

f´(x)=(xctgx)´= (x´)ctgx + x(ctgx)´=ctgx -(x/(sin²x))

f´(π/4)=ctg(π/4) - (π/4)/sin²(π/4)=1 - (π/4)/(√2/2)²=1-(π/4)/(2/4)=

=1-(π/4)/(1/2)=1-(2π/4)=1-(π/2)= (2-π)/2