Differentiate each of the functions with respect to ‘x’

Differentiate using first principle

Let f(x) = x^2/3 ------ (i)

f(x + ∆x) =(x + ∆x)^2/3 ------ (ii)

Subtracting eq. (i) from eq. (ii),

f(x + ∆x) - f(x) = (x + ∆x)^2/3 - x^2/3

Dividing both sides by ∆x and taking the limit,

By differentiation,

Expanding by binomial theorem, and rejecting the higher powers of ∆x as ∆x → 0

Hence, the answer is .