Differentiate : x/(1 - ln(x - 1))

Question

Differentiate : x/(1 - ln(x - 1))

#27, p 223

Answer 1

We have a choice between using the product rule or the quotient rule. I'm going to go with the product rule in this case.

 

Remember that when we have something of the form 1/(.....), we can always rewrite it as (.....)^-1. Any time something is in the denominator, we can bring it to the numerator and negate the sign of its exponent. We'll do that to get:

 

f(x)= x(1-ln[x-1])^-1

 

so now we're looking at x multiplied by a term in brackets. The product rule is:

 

d/dx(g(x)*h(x))= g*h'+h*g'   (using primes for derivatives).

 

We have:

• g(x)=x
• g'(x)=1
• h(x)=(1-ln[x-1])^-1
• h'(x)= (-1)*(1-ln[x-1])^-2*d/dx(1-ln[x-1]) = -(1-ln[x-1])^-2(-1/[x-1]) = -1/[(1-ln[x-1])²*(x-1)]     (via chain rule)

 

Substituting into the product rule, we get the derivative to be

 

{-x/[(1-ln[x-1])2*(x-1)]}  +  {1/(1-ln[x-1])}