Home > Calculus > First Principles Example 3: square root of x

Differentiate the function. H(x) = (x + x^−1)^3 ?

I don't understand how to differentiate this function #H(x)=(x+x^-1)^3# using the binomial theorem. Can someone please go over how to use that theorem to solve this function step by step? Much thanks!

Answer 1

Charles Arocha

# 3x^2+3-3x^{-2}-3x{-4}#

The binomial theorem tells us that

#(a+b)^3=a^3+3a^2b+3ab^2+b^3#

Putting #a=x, b=x^{-1}# gives

#(x+x^{-1})^3=x^3+3x^2x^{-1}+3x (x^{-1})^2 +(x^{-1})^3 = x^3+3x+3x^{-1}+x^{-3}#

This can be easily differentiated using the rule that

# d/{dx} x^n = n x^{ n-1}#

to get

# 3x^2+3-3x^{-2}-3x^{-4}#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.