Could someone help to solve d/dx(ln(e^x(x-1/x+1)^3/2)=?

2 Answers
Feb 24, 2018

d/dx(ln(e^x(x-1/x+1)^(3/2)))=1+(3(2x+1))/(2(x^2+x-1))-3/(2x)

Explanation:

ln(e^x(x-1/x+1)^(3/2))

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

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

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

Hence d/dx(ln(e^x(x-1/x+1)^(3/2)))

= d/dx(x+3/2ln(x^2+x-1)-3/2lnx)

= 1+3/2*1/(x^2+x-1)*(2x+1)-3/2*1/x

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

Feb 24, 2018

\qquad \qquad \qquad \qquad \qquad \quad f'(x) \ = \ { 2 x^3 + 5 x^2 - 2 x + 3 }/( 2 x (x^2 + x - 1 ) ).

Explanation:

"One way to do this conveniently is by using properties of"
"logarithms."

"We want to compute:"

\qquad \qquad \qquad \qquad \qquad \qquad d/{dx} ln( e^x ( x - 1/x + 1)^{3/2} ).

"Let me take the function in question, by itself, first:"

\qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ ln( e^x ( x - 1/x + 1)^{3/2} ).

"Using fundamental properties of logarithms, we can"
"simplify" \ \ f(x) \ \ "greatly:"

\qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ ln( e^x ( x - 1/x + 1)^{3/2} )

\quad = \ ln( e^x ) + ln( x - 1/x + 1)^{3/2}; \qquad \ \ ln( A cdot B ) = ln(A) + ln(B)

\quad = \ x ln( e ) + 3/2 ln( x - 1/x + 1); \quad \ ln( A^r ) = r ln(A)

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ x ln( e ) + 3/2 ln( x - 1/x + 1)

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ x cdot 1 + 3/2 ln( x - 1/x + 1)

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ x + 3/2 ln( x - 1/x + 1).

"So:"

\qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ x + 3/2 ln( x - 1/x + 1).

"Now, finding the derivative is easy [except for simplification !!]:"

\qquad \qquad \qquad \qquad \qquad \quad f'(x) \ = \ [ x ]' + [ 3/2 ln( x - 1/x + 1) ]'

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ ln( x - 1/x + 1) ]'

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( x - 1/x + 1)' ]

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( [ x ]' - [ 1/x ] + 0) ]

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( 1 - [ x^{-1} ]' + 0) ]

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( 1 - [ (-1) x^{-2} ] ) ]

\qquad \qquad "now simplify:"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( 1 - [ (-1) / x^{2} ] ) ]

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 1 + 3/2 [ 1/( x - 1/x + 1)( 1 + 1 / x^{2} ) ]

\qquad \qquad \qquad \qquad= \ 1 + 3/2 [ ( 1/( x - 1/x + 1 ) cdot x/x )( x^2 / x^2 + 1 / x^{2} ) ]

\qquad \qquad \qquad \qquad= \ 1 + 3/2 [ ( color{red}cancel{x}/( x^2 - 1 + x ) )( ( x^2 + 1 ) / x^{ color{red}cancel{2} } ) ]

\qquad \qquad \qquad \qquad= \ 1 + 3/2 [ ( 1/( x^2 - 1 + x ) )( ( x^2 + 1 ) / x ) ]

\qquad \qquad \qquad \qquad= \ 1 + 3/2 [ ( x^2 + 1)/( x (x^2 + x - 1 ) ) ]

\qquad \qquad \qquad \qquad= \ 1 + { 3 ( x^2 + 1) }/( 2 x (x^2 + x - 1 ) )

\qquad \qquad \qquad \qquad= \ ( 2 x (x^2 + x - 1 ) )/( 2 x (x^2 + x - 1 ) ) + { 3 ( x^2 + 1) }/( 2 x (x^2 + x - 1 ) )

\qquad \qquad \qquad \qquad= \ { 2 x (x^2 + x - 1 ) + 3 ( x^2 + 1) }/( 2 x (x^2 + x - 1 ) )

\qquad \qquad \qquad \qquad= \ { 2 x^3 + 2 x^2 - 2 x + 3 x^2 + 3 }/( 2 x (x^2 + x - 1 ) )

\qquad \qquad \qquad \qquad= \ { 2 x^3 + 5 x^2 - 2 x + 3 }/( 2 x (x^2 + x - 1 ) ).

"Sorry for all the work -- the calculus part was shorter,"
"the simplication long (not uncommon) !!"

"Summarizing:"

\qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ ln( e^x ( x - 1/x + 1)^{3/2} ).

\qquad \qquad \qquad \qquad \qquad \quad f'(x) \ = \ { 2 x^3 + 5 x^2 - 2 x + 3 }/( 2 x (x^2 + x - 1 ) ).