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

1 Answer
Shwetank Mauria
Mar 20, 2018

Derivative is 3(x+1/x)^2(1-1/x^2)3(x+1x)2(1−1x2)

Explanation:

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

Now according to chain formula (df)/(dx)=(df)/(dg)*(dg)/(dx)dfdx=dfdg⋅dgdx

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

and as g(x)=x+1/xg(x)=x+1x, we have (dg)/(dx)=1-1/x^2dgdx=1−1x2

Hence (df)/(dx)=3(g(x))^2*(dg)/(dx)=3(x+1/x)^2(1-1/x^2)dfdx=3(g(x))2⋅dgdx=3(x+1x)2(1−1x2)

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