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Question
`x^(2/3)`
Solution
Let `f(x) = x^(2/3)` ....(i)
`f(x + Δx) = (x + Δx)^(2/3)` .....(ii)
Subtracting equation (i) from (ii) we get
`f(x + Δx) - f(x) = (x + Δx)^(2/3) - x^(2/3)`
Dividing both sides by Δx and take the limit.
`lim_(Δx -> 0) (f(x + Δx) - f(x))/(Δx) = lim_(Δx -> 0) ((x + Δx)^(2/3) - x^(2/3))/(Δx)`
f'(x) = `lim_(Δx -> 0) (x^(2/3) [1 + (Δx)/x]^(2/3) - x^(2/3))/(Δx)` ........[By definition of differentiation]
= `lim_(Δx -> 0) (x^(2/3) [(1 + (Δx)/x)^(2/3) - 1])/(Δx)`
= `lim_(Δx -> 0) (x^(2/3) [(1 + 2/3 * (Δx)/x + ...) - 1])/(Δx)`
[Expanding by Binomial theorem and rejecting the higher powers of Δx as Δx → 0]
= `lim_(Δx -> 0) (x^(2/3) * 2/3 * (Δx)/x)/(Δx)`
= `2/3 x^(2/3 - 1)`
= `2/3 x^((-1)/3)`
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