Question

Найти производные функций при данном значении аргумента
1.f(z) = ln sin z; f ' (pi/6)
2. F(x) = cos x + sin^2x; F'(0)
3. f(x)= ln ctg x; f ' (pi/4)
4. f(y) = e ^ cos2y; f ' (pi/4)

Answer 1

1. f(z) = ln(sinz); f'(pi/6)

f'(z) = 1/sinz * cosz = ctg(z)

f'(pi/6) = ctg(pi/6) = √3

2. F(x) = cos x + sin^2x; F'(0)

F'(x) = -sinx + 2sinxcosx

F'(0) = 0

3. f(x) = ln(ctgx); f'(pi/4)

f'(x) = 1/ctgx * -1/sin^2x = -1/(sinx*cosx)

f'(pi/4) = -2

4. f(y) = e^cos2y; f'(pi/4)

f'y = e^cos2y * -sin2y * 2 = -2sin2y * e^cos2y

f'(pi/4) = -2