Question

Вычислить производную первого и второго порядка: y=(1-ctg(x)/3)^5

Answer 1

$y=\Big(1-ctg\dfrac{x}{3}\Big)^5\ \ \ ,\ \ \ \ (u^5)'=5u^4\cdot u'\\\\\\y'=5\Big(1-ctg\dfrac{x}{3}\Big)^4\cdot \Big(-\dfrac{-1}{sin^2\frac{x}{3}}\Big)\cdot \dfrac{1}{3}=5\Big(1-ctg\dfrac{x}{3}\Big)^4\cdot \dfrac{1}{3\, sin^2\frac{x}{3}}\\\\\\y''=5\cdot 4\Big(1-ctg\dfrac{x}{3}\Big)^3\cdot \dfrac{1}{3\, sin^2\frac{x}{3}}\cdot \dfrac{1}{3sin^2\frac{x}{3}}+5\Big(1-ctg\dfrac{x}{3}\Big)^4\cdot \dfrac{-6sin\frac{x}{3}\cdot cos\frac{x}{3}\cdot \frac{1}{3}}{9sin^4\frac{x}{3}}=$

$=20\cdot \Big(1-ctg\dfrac{x}{3}\Big)^3\cdot \dfrac{1}{9sin^4\frac{x}{3}}-5\Big(1-ctg\dfrac{x}{3}\Big)^4\cdot \dfrac{2\cdot cos\frac{x}{3}}{9sin^3\frac{x}{3}}$