logarithmic differentiation to find the derivative of the equationy=(sqrt(x))^(5x)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use logarithmic differentiation, hence, you need to take natural logarithm to both sides such that:

`ln y = ln (sqrt(x))^(5x)`

You need to use properties of logarithms such that:

`ln y = 5x*ln (sqrt(x))`

You need to differentiate both sides, using the product rule to the right such that:

`(y')/y = (5x)'*ln (sqrt(x)) + 5x*(ln (sqrt(x)))'`

`(y')/y = 5*ln (sqrt(x)) + 5x*((sqrt x)')/(sqrt x)`

`(y')/y = 5*ln (sqrt(x)) + (5x)/(2(sqrt x)(sqrt x))`

`(y')/y = 5*ln (sqrt(x)) + (5x)/(2x)`

`(y')/y = 5*ln (sqrt(x)) + 5/2`

`(y')/y = 5*(ln (sqrt(x)) + 1/2)`

You need to multiply by `y`  both sides such that:

`y' = 5y*(ln (sqrt(x)) + 1/2)`

You need to substitute `(sqrt(x))^(5x)`  for `y`  such that:

`y' = 5(sqrt(x))^(5x)*(ln (sqrt(x)) + 1/2)`

Hence, using logarithmic differentiation yields `y' = 5(sqrt(x))^(5x)*(ln (sqrt(x)) + 1/2) .`