How to solve this derivative?

Derivative of lnx is 1/x.

Use the quotient rule.

Or you can change it to (1+e^x)^-1 and use the product rule.

Either way, that is not fun. Good luck.

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Note:

d/dx(lnu) = (1/u)(du/dx)

So,

d/dx(lnx) = (1/x)(dx/dx) = 1/x

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f(x) = (ln(x)) / (1 + eˣ)                       ← I'll differentiate using the quotient rule

           (1 + eˣ)(1/x) - (ln(x))(0+eˣ)

f '(x) = —————————————–

                         (1 + eˣ)²

           x[(1 + eˣ)(1/x) - (ln(x))(0+eˣ)]

f '(x) = —————————————–—        ← Multiplied top and bottom by x to eliminate 1/x

                         x(1 + eˣ)²

           1 + eˣ - xeˣ(ln(x))

f '(x) = ——————————            ← ANSWER ... 1st derivative

                  x(1 + eˣ)²

http://www.wolframalpha.com/input/?i=differentiate...

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Now, use the quotient rule again to get the 2nd derivative.

Note first that:

d/dx(uvw) = (d/dx(u)·vw) + (d/dx(v)·uw) + (d/dx(w)·uv)

So,

   d xeˣ ln x

  ————— = (eˣ ln x)(dx/dx) + (x ln x)(d/dx(eˣ)) + (xeˣ)(d/dx(ln x))

         dx

 d/dx(xeˣ ln x) = (eˣ ln x) + (x ln x) eˣ + (xeˣ)(1/x)

 d/dx(xeˣ ln x) = eˣ ln x   +   eˣ x ln x + eˣ

 d/dx(xeˣ ln x) = (ln x + x ln x + 1) eˣ

 d/dx(xeˣ ln x) = [(1 + x)ln x + 1] eˣ            ← Now, we can substitute this as needed

           d 1+eˣ–xeˣ ln x

f ''(x) = ————————            ← Now, differentiate using the quotient rule

                 x(1 + eˣ)²

           x(1 + eˣ)²•(0+eˣ-[(1+x)ln x + 1] eˣ) − (1+eˣ–xeˣ ln x)•[x · 2(1+eˣ)eˣ + (1 + eˣ)²·1]

f ''(x) = ———————————————————————————————————

                                                         [ x(1 + eˣ)² ]²

           -xeˣ(1 + eˣ)²(1+x)ln x − (1+eˣ–xeˣ ln x)(1+eˣ)(2xeˣ+eˣ+1)

f ''(x) = ——————————————————————————   ← divide top and bottom by (1+eˣ)

                                              x²(1 + eˣ)⁴

           -xeˣ(1 + eˣ)(1+x)ln x − (1+eˣ–xeˣ ln x)(2xeˣ+eˣ+1)      ← Expand this and collect like terms

f ''(x) = ———————————————————————

                                          x²(1 + eˣ)³

           eˣ(eˣ - 1)x² ln x - (eˣ + 1)(eˣ(2x+1) + 1)

f ''(x) = ——————————————————            ← ANSWER ... 2nd derivative

                                  x²(eˣ + 1)³

http://www.wolframalpha.com/input/?i=differentiate...

Hope that is easy enough to follow.

Have a good one!

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