Question

Найти производные функций, во вложении.

Answer 1

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Answer 2

1.
$y= frac{lnx}{sinx}+x*ctgx \ y'= frac{ frac{sinx}{x}-lnx*cosx}{sin^2x}+ctgx- frac{x}{sin^2x} = frac{sinx-x*lnx*cosx}{x*sin^2x}+ frac{cosx}{sinx}- frac{x}{sin^2x}=$$frac{sinx-x*lnx*cosx+x*sinx*cosx-x^2}{xsin^2x} =frac{sinx-x^2+x*cosx(sinx-lnx)}{xsin^2x}$
2.
$y= frac{ sqrt{x}}{sqrt{x}+1} \y'= frac{ frac{ sqrt{x}+1}{2sqrt{x}}-frac{ sqrt{x}}{2sqrt{x}}}{(sqrt{x}+1)^2}= frac{1}{2sqrt{x}*(sqrt{x}+1)^2}$
3.
$y=frac{ctgx}{ sqrt{x}} \ y'= frac{- frac{ sqrt{x}}{sin^2x}- frac{ctgx}{2 sqrt{x} } }{x}=- frac{2x+sinx*cosx}{2x sqrt{x} *sin^2x} =- frac{4x+2sinx*cosx}{4x sqrt{x} *sin^2x} =- frac{4x+sin2x}{4x sqrt{x} *sin^2x}$
4.
y = ln (x² + 2x)
$y'= frac{2x+2}{x^2+2x}= frac{2(x+1)}{x(x+2)}$
5.
$y=ln frac{x^2}{1-x^2} \ y'=frac{1-x^2}{x^2}*frac{2x(1-x^2)+x^2*2x}{(1-x^2)^2}=frac{2}{x(1-x^2)}$