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Ответ

Ответ дан
Андрей Панарин
Производные:
(arctg x)' = 1 / (1 + x²)
(arcctg x)' = −1 / (1 + x²)

361. y(x) = arctg x³
Введем z(x) = x³; z'(x) = 3x²
Тогда y(z) = arctg z
y'(z) = 1 / (1 + z²)

y'(x) = dy / dx = dy/dz • dz/dx = y'(z) • z'(x) =
= (arctg z)' • (x³)' = 1 / (1 + z²) • 3x² = 3x² / (1 + (x³)²) = 3x² / (1 + x^6)

По этому же шаблону можно решить номера 360 и 364.

Номер 371 решается через производную частного:
(u / v)' = (u'v − uv') / v²,
где u = x², v = arctg x.

В номере 372 тоже вводим новые функции:
a(x) = (1 − x) / (1 + x)
b(a) = Ѵa
y(a) = arcctg b
И узнаем их производные.

А затем y'(x) = dy / dx = dy/db • db/da • da/dx = y'(b) • b'(a) • a'(x) = …

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