derivatives

Question

Find dy/dx when y=x^(4lnx) 

Answer 1

y = X^{4ln x}

Apply ln on each side .

ln y = ln (X^4ln x) ⇒ 4 ln x ( ln x )                          [Since ln X^a = a ln x]

ln y =4( ln x )²  

Now apply derivative with respect to x.

(1/y) dy/dx = 4 * 2 (ln x) [ln x]'                           [ derivative of  xⁿ = n x^n-1 and ln x = 1/x ]        

(1/y) dy/dx = 8 (ln x) [1/ x]

Multiply each side by y.

 dy/dx = 8 (ln x) [1/ x] *y

 dy/dx = 8 (ln x) [1/ x] * X^4ln x                                      [ Since y = X^4ln x ]

 dy/dx = 8 (ln x) [X ^{- 1}] * X^4ln x

So  dy/dx = 8 (ln x)  X^{4ln x -1}