Suppose we are interested in the condition number of a monomial, not just one monomial, but all monomials…

$$ f(x) = b x^n $$

where

• $b \in \mathbb{R}$
• $x \in \mathbb{R}$
• $n \ne 0$

We will assume the general form of the condition number, i.e., we will not address $x = 0$ or $f(x) = 0$.

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Computing the condition number is fairly quick… once we recall that

$$ \begin{eqnarray} f^\prime(x) &=& \frac{d}{dx}(b x^n)\\ &=& bnx^{n-1}\\ \end{eqnarray} $$

Recall the various properties of exponents

Do not forget that $x^n * x^m = x^{n+m}$.

The condition number can be computed fairly quickly…

$$ \begin{eqnarray} (cond\phantom{1}f)(x) &=& \left| \frac{xf^\prime(x)}{f(x)} \right|\\ &=& \left| \frac{x * bnx^{n-1}}{bx^n}\right|\\ &=& \left| \frac{x * x^{n-1}}{x^n}\right|\\ &=& \left| \frac{nx^{n}}{x^n}\right|\\ (cond\phantom{1}f)(x) &=& \left| n \right|\\ \end{eqnarray} $$

Interesting… does the result make sense? This result indicates that the power (exponent) of a monomial determines its conditioning. This is a useful result to keep in mind.