find derivative---

Question

f(x)=lnx/x²

Answer 1

F (x) = x^-2 ln x

 

Finally we differentiate f (x) by using the product rule

 

Our u = x^-2 and v = ln x

 

Du / dx = -2x^-3

 

dv/dx = 1 / x 

 

Finally our derivative is

 

 X^-2 ( 1/x) + ln x (-2x^-3)

 

= 1 / x^3 - (2 ln x ) / x^3

 

 

 

 

 

Answer 2

d/dx (ln x)/x²

 

= [x² d/dx ln x - (ln x) d/dx x²] / (x²)²

 

= [x²(1/x) - 2x ln x] / x⁴

 

=  (x - 2x ln x) / x⁴

 

= (1 - 2 ln x) / x³

Answer 3

let f(x) = x^-2ln x = uv  with u' = -2x^-3 and v' = x^-1

(uv)' = u'v + uv'  [product rule]  so f'(x) = -2x^-3ln x + x^-2(x^-1) = x^-3(1-2ln x)