Differentiate Ln ( SQRT(x-2))/X²

Question

Differentiate  Ln  od  square root of x-2  divided  by  x  squared.

 

Thanks  You

Regards

Sunny 

Answer 1

Is the x² inside or outside of the ln? It does make a difference, so I will show you both ways.

 

1)  y = ln {[√(x - 2)]/x²} = ½ ln (x - 2) - 2 ln x     (expand using properties of logarithms)

 

     dy/dx = ½[1/(x - 2)] - 2(1/x) = 1/[2(x - 2)] - 2/x

 

2)  y = [ln (√(x - 2)]/x² = [ln (x - 2)]/(2x²)     (simplify using properties of logarithms)

 

     dy/dx = [2x²/(x - 2) - 4x ln (x - 2)]/(4x⁴)    (Quotient Rule)

 

              = [x - (2x - 4) ln (x - 2)]/[2x³(x - 2)]    (reducing the fraction)

 

Sorry I had to do this twice, but you must be very careful about parentheses when submitting problems which can be interpreted in more than one way. Perhaps I am being too cautious, but many students forget to do this.

 

Let me know if this helps you.

Answer 2

    √(x-2)
ln --------- is best simplified as  y = ½ln(x-2) - 2lnx
      x²

 

and then y' =½(1/(x-2)) -2(1/x) and if that is put over the common denominator  2x(x-2) the numerator becomes

 

x - 2•(x-2)  and I get the final answer (4 - x) / [2x(x-2)]

so I am not in agreement with Marlene's answer.

 

R.S.V.P.

Answer 3

Thanks Sunny.

 

OK.  There are a few differentiation rule in play here

 

1) d/dx lnx = 1/x

2) ln x/y = lnx - lny

3) chain rule

 

We have

     √(x-2)

ln ---------

        x²

 

Let's rewrite that as

ln √(x-2) - ln x²

and

ln (x-2)^1/2 - ln x²

 

First take the derivative of ln which is

1/(x-2)^1/2 d/dx (x-2)^1/2 - 1/x² d/dx(x²)

 

continuing

 

1/(x-2)^1/2 (1/2)(x-2)^-1/2 - 1/x²(2x)

 

Simplifying

 

The first term reduces to 1/2 and the second term reduces to 2/x so we have

 

1/2 + 2/x 

 

(x + 4)/2x