Which function is the antiderivative for the function y = 7x ^ 6-15x ^ 4?

Using basic differentiation formulas and differentiation rules:

(x ^ n) ‘= n * x ^ (n-1).

(c) ‘= 0, where c is const.

(c * u) ’= c * u’, where c is const.

(u ± v) ‘= u’ ± v ‘.

(uv) ‘= u’v + uv’.

y = f (g (x)), y ‘= f’u (u) * g’x (x), where u = g (x).

Thus, the derivative of our given function will look like this:

f (x) ‘= ((x ^ 2 + 6x + 12) ^ (1/2))’ = ((x ^ 2 + 6x + 12)) ‘* ((x ^ 2 + 6x + 12) ^ ( 1/2)) ‘= ((x ^ 2)’ + (6x) ‘+ (12)’) * ((x ^ 2 + 6x + 12) ^ (1/2)) ‘= (2x + 6 + 0) * (1/2) * ((x ^ 2 + 6x + 12) ^ (- 1/2)) = (x + 3) / ((x ^ 2 + 6x + 12) ^ (1/2) ).

Answer: The derivative of our given function will be equal to f (x) ‘= (x + 3) / ((x ^ 2 + 6x + 12) ^ (1/2)).