. Найдите производную функции: а) у=х4; б) у=4 г) у=3х + 2 е) y=x sinx

а) f(x)' = (x * (x – 4))’ = (x)’ * (x – 4) + x * (x – 4)’ = (x)’ * (x – 4) + x * ((x)’ – (4)’) = 1 * (x – 4) + x * (1 – 0) = x – 4 + x = 2x – 4.

б) f(x)' = (3 – 2х)’ = (3)’ – (2х)’ = 0 – 2 * 1 * x^{(1 – 1)} = -2 * x⁰ = -2 * 1 = -2.

г) f(x)' = ((-x^2 + 2x^2)^3 + (x - 3)^4)’ = ((-x^2 + 2x^2)^3)’ + ((x - 3)^4)’ = (-x^2 + 2x^2)’ * ((-x^2 + 2x^2)^3)’ + (x - 3)’ * ((x - 3)^4)’ = ((-x^2)’ + (2x^2)’) * ((-x^2 + 2x^2)^3)’ + ((x)’ – (3)’) * ((x - 3)^4)’ = (-2x + 4x) * (3 * (-x^2 + 2x^2)^2) + (1 - 0) * (4 * (x - 3)^3) = 2x * 3 * (x^2)^2 + 1 * (4 * (x - 3)^3) = 6x * x^4 + 4(x - 3)^3 = 6x^5 + 4(x - 3)^3.

е) f(x)' = (2sin (x))’ = 2 * (sin (x))’ = 2 * cos (x) = 2cos (x).

з) f(x)' = ((3^x) * (x + 1))’ = ((3^x)’ * (x + 1)) + ((3^x) * (x + 1)’) = ((3^x)’ * (x + 1)) + ((3^x) * ((x)’ + (1)’) = (3^x * ln 3) * (x + 1)) + (3^x) * 1 = (3^x * ln 3) * (x + 1)) + (3^x).