Find the derivative of the function

;

Do i have to use logarithmic differentiation?

The answer according to the back of book is....

how?

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- Jan 27th 2007, 05:01 PM^_^Engineer_Adam^_^Logarithmic Diff.....
Find the derivative of the function

;

Do i have to use logarithmic differentiation?

The answer according to the back of book is....

how? - Jan 27th 2007, 05:13 PMThePerfectHacker
- Jan 27th 2007, 10:08 PMCaptainBlack
- Jan 28th 2007, 12:45 AMticbol
Here is one way.

f(x) = x^(sqrt(x))

Let y = f(x)

So,

y = x^(sqrt(x))

Take the ln of both sides,

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

Differentiate both sides with respect to x,

(1/y)y' = [sqrt(x) *1/x] +[ln(x) *(1/2)x^(-1/2)]

(1/y)y' = [1/sqrt(x)] +[1/(2sqrt(x)) *ln(x)]

Factor the RHS by 1/sqrt(x),

(1/y)y' = (1/sqrt(x))[1 +(1/2)ln(x)]

Multiply both sides by y,

And since 1/sqrt(x) = x^(-1/2),

y' = (x^sqrt(x))*(x^(-1/2))[1 +(1/2)ln(x)]

y' = [x^(sqrt(x) -(1/2)][1 +(1/2)ln(x)] -----answer at the back of the book.