Вычислите производную в точке. Created by muuuuuuuuuuuur. algebra-ru

Вычислите производную в точке...

Minsk00 Вычислите производную в точке g(x)=4ctg(x) g'(-2/3)
Решение
g'(x) =(4ctg(x))' = -4/sin²(x)
sin(-2/3) = -sin(2/3) = -sin(2*180/3)= -sin(120) =-sin(90+30) =-cos(30) = -√(3)/2
g'(-2/3) = -4/(-√(3)/2)² =-4/(3/4) = -16/3 = -5 1/3 ≈ -5,33
Ответ: -16/3

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мохинсан $g(x)=4ctgx;\\
g'(-\frac{5\pi}{3})-?;\\
g'(x)=(4ctgx)'=4(ctgx)'=4(\frac{\cos x}{\sin x})'=4\frac{(\cos x)'\cdot\sin x-\cos x\cdot(\sin x)'}{\sin^2x}=\\
=4\frac{-\sin x\cdot\sinx-\cos x\cdot\cos x}{\sin^2x}=-4\frac{\sin^2x+\cos^2x}{\sin^2x}=\frac{-4}{\sin^2x};\\
g'(-\frac{2\pi}{3})=\frac{-4}{\sin^2(-\frac{2\pi}{3})}=\frac{-4}{(\frac{\sqrt{3}}{2})^2}=\frac{-4}{\frac{3}{4}}=-\frac{16}{3}=-5\frac{1}{3}$

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