Use logarithimic differentiation to find the derivative of y.

y= (x(√(x^5)+8))/((x-3)^(2/3))

First, you must realize that logarithmic differentiation is the most efficient and least time consuming way to find dy/dx. Using the product rule or the quotient rule to find the derivative would take way too long and it would be rather tedious.

ln(y) = ln(x) + 1/2*ln(x^5 + 8)) - 2/3*ln(x - 3)

Differentiating both sides:

y'/y = 1/x + (5x^4)/(2x^5 + 16) - 2/3*1/(x - 3)

y' = [(x(√(x^5)+8))/((x-3)^(2/3))] * [1/x + (5x^4)/(2x^5 + 16) - 2/3*1/(x - 3)]

y = x(√(x^5)+8)/((x-3)^(2/3))

ln y = ln (x(√(x^5)+8)) - (2/3) ln (x - 3)

implicit differentiation,

(1/y) y' = 1/(x(√(x^5)+8)) * ( (7 √(x^5))/2 + 8) - 2/(3(x - 3)) * 1

y' = [1/(x(√(x^5)+8)) * ( (7 √(x^5))/2 + 8) - 2/(3(x - 3))] (x(√(x^5)+8)/((x-3)^(2/3)))

y' = [1/(x(√(x^5)+8)) * ( (7 √(x^5))/2 + 8) - 2/(3(x - 3))] y

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