Question

How do you differentiate `(1+ (1/x))^3`?

Answer 1

Derivative is `3(x+1/x)^2(1-1/x^2)`

Explanation:

We use the concept of function of a function. Let `f(x)=(g(x))^3`, where `g(x)=1+1/x` and then `f(x)=(1+1/x)^3`

Now according to chain formula `(df)/(dx)=(df)/(dg)*(dg)/(dx)`

As `f(x)=(g(x))^3`, `(df)/(dg)=3(g(x))^2`

and as `g(x)=x+1/x`, we have `(dg)/(dx)=1-1/x^2`

Hence `(df)/(dx)=3(g(x))^2*(dg)/(dx)=3(x+1/x)^2(1-1/x^2)`