Question

Найти производные 1) y=x^3-5/(x^2+1) 2) y=e^2xx^3 3)y= sin(4x-8)

Answer 1

$(f(x))' = f'(x)*x'\\1) y = x^3 - \frac5{x^2+1}\\(f(x) - g(x))' = f'(x) - g'(x)\\ (\frac1x)' = -\frac1 {x^2} * x'\\ (x^n)' = nx^{n-1}\\y' = 3x^2 + \frac{5*(x^2+1)'}{(x^2+1)^2}=3x^2 + \frac{10x}{(x^2+1)^2}\\2) y = e^{2x}x^3\\(f(x)*g(x))' = f'(x)g(x) + f(x)g'(x)\\(e^x)' = e^x*x'\\y' = x^3e^{2x}*(2x)' + (x^3)'e^{2x} = 2x^3e^{2x} + 3x^2e^{2x}\\3) y = sin(4x -8)\\(sinx)' = cosx*x'\\y' = (4x-8)'cos(4x-8) = 4cos(4x-8)$