Find the derivative of the functions y = (x ^ 4-3x ^ 3) ^ 42

Let’s find the derivative of our given function: f (x) = (x ^ 4 – 3x ^ 3) ^ 42.

Using the 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 ‘.

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

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

f (x) ‘= ((x ^ 4 – 3x ^ 3) ^ 42)’ = (x ^ 4 – 3x ^ 3) ‘* ((x ^ 4 – 3x ^ 3) ^ 42)’ = ((x ^ 4) ‘- (3x ^ 3)’) * ((x ^ 4 – 3x ^ 3) ^ 42) ‘= (4 * x ^ 3 – 3 * 3 * x ^ 2) * 42 * (x ^ 4 – 3x ^ 3) ^ 41 = 42 * (4x ^ 3 – 9x ^ 2) * (x ^ 4 – 3x ^ 3) ^ 41.

Answer: The derivative of our given function will be equal to f (x) ‘= 42 * (4x ^ 3 – 9x ^ 2) * (x ^ 4 – 3x ^ 3) ^ 41.