Find the derivative of each function y = x + 5x ^ 6 + 1.

Let’s find the derivative of our given function: f (x) = 5x ^ 6 + x + 1.

Let’s use the basic rules and formulas for differentiation:

(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).

That is, the derivative of our given function will be as follows:

f (x) ‘= (5x ^ 6 + x + 1)’ = (5x ^ 6) ‘+ (x) + (1)’ = 5 * 6 * x ^ (6 – 1) + 1 * x ^ ( 1 – 1) + 0 = 30x ^ 5 + 1.

Answer: The derivative of our given function f (x) ‘= 30x ^ 5 + 1.