Найти производные функций, заданных в явном и неявном виде. а) y=(1+(x-1)/(x+1))^sqrt(2+x^2); б) x^3y-xy^3=sin(xy)

f(x)\' = ((-1 / 5) *x^3 + 45x^2 + 4x – 1)’ = ((-1 / 5) * x^3)’ + (45x^2)’ + (4x)’ – (1)’ =

(-1 / 5) * 3 * x^2 + 45 * 2 * x + 4 – 0 = (-3 / 5) * x^2 + 90x + 4.

y\' = ((cos^2 x) * (1 / x))’ = (cos^2 x)’ * (1 / x) + (cos^2 x) * (1 / x)’ = (cos x)’ * (cos^2 x)’ * (1 / x) + (cos^2 x) * (1 / x)’ = (-2 * (sin x) * (cos x) * (1 / x)) + (cos^2 x) * (-1 / x^2) = (-2(sin x)(cos x) / x) + (cos^2 x / x^2).

f(x)\' = ((x^2 + 1)^-(1 / 2))’ = (x^2 + 1)’ * ((x^2 + 1)^(-1 / 2))’ = ((x^2)’ + (1)’) * ((x^2 + 1)^(-1 / 2))’ = 2x * (1 / 2) * (x^2 + 1)^(-3 / 2) = x / ((x^2 + 1)^(3 / 2)) = x / √((x^2 + 1)^3).

y\' = (x^2 * ln (tg √2x))’ = (x^2)’ * ln (tg √2x) + x^2 * (ln (tg √2x))’ = (x^2)’ * ln (tg √2x) + x^2 * (tg √2x)’ * (ln (tg √2x))’ = (x^2)’ * ln (tg √2x) + x^2 * (√2x)’ * (tg √2x)’ * (ln (tg √2x))’ = 2 * x^(2 – 1) * ln (tg √2x) + x^2 * √2 * (1 / (cos^2 √2x) * (1 / (tg √2x)) = 2xln (tg √2x) + (√2x^2 / (cos^2 √2x)(tg √2x)).