Differentiate y=`(sqrt x*(1 - 2x)^(2/3))/((2-3x)^(3/4)*(3-4x)^(4/5))`

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The function y = `(sqrt x*(1 - 2x)^(2/3))/((2-3x)^(3/4)*(3-4x)^(4/5))`

=> y = `x^(1/2)*(1 - 2x)^(2/3)*(2 - 3x)^(-3/4)*(3-4x)^(-4/5)`

Use the product rule to find the derivative of y

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

` x^(1/2)*((1 - 2x)^(2/3))'*(2 - 3x)^(-3/4)*(3-4x)^(-4/5) + `

`x^(1/2)*(1 - 2x)^(2/3)*((2 - 3x)^(-3/4))'*(3-4x)^(-4/5)+ `

`x^(1/2)*(1 - 2x)^(2/3)*(2 - 3x)^(-3/4)*((3-4x)^(-4/5))'`

=>` (1/(2*sqrt x))*(1 - 2x)^(2/3)*(2 - 3x)^(-3/4)*(3-4x)^(-4/5) + `

`x^(1/2)*(-4/3)*(1 - 2x)^(-1/3)*(2 - 3x)^(-3/4)*(3-4x)^(-4/5) + `

`x^(1/2)*(1 - 2x)^(2/3)*(-9/4)*(2 - 3x)^(-7/4)*(3-4x)^(-4/5)+ `

`x^(1/2)*(1 - 2x)^(2/3)*(2 - 3x)^(-3/4)*(16/5)*(3-4x)^(-9/5)`

=>

`((1 - 2x)^(2/3))/(2*sqrt x*(2 - 3x)^(3/4)*(3-4x)^(4/5)) -(4*sqrt x)/((1 - 2x)^(1/3)*(2 - 3x)^(3/4)*(3-4x)^(4/5))`

`- (9*sqrt x*(1 - 2x)^(2/3))/(4*(2 - 3x)^(7/4)*(3-4x)^(4/5))+(16*sqrt x*(1 - 2x)^(2/3))/(5*(2 - 3x)^(3/4)*(3-4x)^(9/5)) `

**The required derivative is** `((1 - 2x)^(2/3))/(2*sqrt x*(2 - 3x)^(3/4)*(3-4x)^(4/5)) -(4*sqrt x)/((1 - 2x)^(1/3)*(2 - 3x)^(3/4)*(3-4x)^(4/5))``- (9*sqrt x*(1 - 2x)^(2/3))/(4*(2 - 3x)^(7/4)*(3-4x)^(4/5))+(16*sqrt x*(1 - 2x)^(2/3))/(5*(2 - 3x)^(3/4)*(3-4x)^(9/5)) `

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