Найти производную и первообразную функций: а) f(x) = x^2 + 3 б) f(x) = cosx + 1/2x^2 в) f(x) = 5x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1 ...

а) f(x) = x^2 + 3 [latex]f'(x) = (x^2)' + (3 )'=2x+0 \\ f'(x) = 2x[/latex] [latex]F(x) = \frac{x^3}{3} + 3 x+C[/latex] б) f(x) = cosx + 1/2x^2 [latex]f'(x) = (cosx)' + \frac{1}{2} (x^2 )'=-sinx+ \frac{1}{2} *2x=-sinx+ x[/latex] [latex]F(x) =sinx + \frac{1}{2} * \frac{x^3}{3} +C=sinx + \frac{x^3}{6}+C[/latex] в) f(x) = 5x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1 [latex]f'(x) = 5(x^5)' + 4(x^4)' + 3(x^3)' + 2(x^2)' + (x)' + (1)'= \\ \\ =5*5x^4 + 4*4x^3 + 3*3x^2 + 2*2x + 1 + 0= \\ \\ =25x^4 + 16x^3 + 9x^2 + 4x + 1[/latex] [latex]F(x) = 5* \frac{x^6}{6} + 4* \frac{x^5}{5} + 3* \frac{x^4}{4} + 2* \frac{x^3}{3} + \frac{x^2}{2} + 1*x+C= \\ \\ =\frac{5x^6}{6}+\frac{4x^5}{5}+\frac{3x^4}{4}+\frac{2x^3}{3}+\frac{x^2}{2}+x+C[/latex]