Derivatives of functions

Question

Find the derivative of the function.

y=(1/2)((1/2) ln (x+1)/(x-1) + arctan x)

Answer 1

We can write this so that it contains no denominators.

 

y = (1/2)[(1/2)(x - 1)^-1ln(x + 1) + arctan(x)]

 

 

Let's distribute the 1/2 to each term in parenthesis to make the derivative easier to find.

 

y = (1/4)(x - 1)^-1ln(x + 1)+ (1/2)arctan(x)

 

 

Now you can derive.  Use a combination of product rule, chain rule, and logarithmic differentiation on the first factor of the function.  Use a combination of product rule and trigonometric differentiation on the second term.  Then add those sub-derivatives together to get the actual derivative.

Answer 2

y = (1/2)[(1/2)ln{(x+1)/(x-1)} + arctanx]

 

   = (1/2)[(1/2)(ln(x+1)-ln(x-1))+arctanx]

 

y' = (1/2)[(1/2)(1/(x+1)-1/(x-1))+1/(1+x²)]

 

   = (1/2)[(1/2)(-2/(x²-1)+1/(1+x²)]

 

   = (1/2)[-1/(x²-1) + 1/(x²+1)]

 

   = (1/2)[(-x²-1+x²-1)/(x⁴-1)]

 

   = (1/2)[-2/(x⁴-1)] = -1/(x⁴-1)