How Does Logarithmic Differentiation Work with Complex Functions?

[helpm3pl3ase]

1. ln x^(1/5)

= 1/5 ln x which = 1/5 *1/x

so overall it = 1/5x correct??

Iam so lost on this problem.

2. F(y) = y ln (1 + e^y)

any help would be appreciated

 

[neutrino]

The first answer is correct. For the second, use the product rule.

 

[helpm3pl3ase]

Alright thank you.. I have one more question though or some work to check to see if I did this correctly. Its on implicit differentiation:

(x+y)^(1/2) = 1 + x^(2)y^(2)

= 1/2 (x+y)^(-1/2) (1 + dy/dx) = x^(2) (2y dy/dx) + y^(2) (2x)

= 1/2 (x+y)^(-1/2) - x^(2) - y^(2) (2x) = -(1 + dy/dx) + (2y dy/dx)

I can get to this point but I don't know how to simplify to get me dy/dx.

Iam not really sure how to get the dy/dx out of this -(1 + dy/dx) because its one term. Any help is apperciated. Thanks

 

[neutrino]

1/2 (x+y)^(-1/2) (1 + dy/dx) = x^(2) (2y dy/dx) + y^(2) (2x)

Expand the term on the left, and then gather all terms containing dy/dx.