What are the derivatives of the cube root of 6x^2?

Answer 1

f (x) = ( 6 x ² )^(1/3)
f `(x) = (1/3) (6 x²) ^(- 2/3)(12 x)
f `(x) = (4x) / ( 6^(2/3) (x^(4/3))
f `(x) = (4) x^(-1/3) / 6^(2/3)
f `(x) = (4) / [ 6^(2/3) x^(1/3) ]

Answer 2

This is a use of the index laws (a^r * b^m) ^ n = a^ (r*n) * b^ (m*n)

f ( x ) = (6x^2)^(1/3) = 6^(1/3) * x^(2/3)
f ' (x) = 6^(1/3) * [(2/3)x^( - 1/3)]

Hope this helps

Answer 3

Rewrite the problem as

f (x) = (6x^2)^(1/3)

Power rule with Chain rule:
d/dx [g(x)]^n = n[g(x)] ^ (n-1) * g ' (x)

f ' (x) = (1/3) * [ (6x^2) ] ^ (-2/3) * 12x or

f ' (x) = 12x / [ 3 * (6x^2) ^(2/3)]