1 Two Methods

1.1 Compute f’(x)

We can compute the derivative of

$$ f(x) = \alpha e^{bx} $$

directly using the exponent rule.

$$ f^\prime(x) = \alpha b e^{bx} $$

1.2 Natural Log Trick

Sometimes… the natural log trick can simplify the algebra. The natural log trick used the derivative of the natural log, i.e.,

$$ \frac{d}{dx}\ln(f(x)) = \frac{f^\prime(x)}{f(x)} $$

Let use use the natural log trick for $f(x) = \alpha e^{bx}$:

$$ \begin{eqnarray} \frac{d}{dx}\ln(\alpha e^{bx}) &=& \frac{d}{dx}\left(\ln(\alpha) + \ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(\ln(\alpha)\right) + \frac{d}{dx}\left(\ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(\ln(e^{bx})\right) \\ &=& \frac{d}{dx}\left(bx\right) \\ &=& b \\ \end{eqnarray} $$

The natural log trick often allows a problem to be simplified algebraically (through properties of the natural log) before computing any deviates.

2 Computing the Condition Number

Now… we can plug the result into the condition number formula.

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

3 Examining Behavior

Suppose we are interested in three cases:

• $x \to 0$
• $x \to \infty$
• $x \to -\infty$

As $x$ approaches 0.. the condition number also approaches zero.

$$ \lim_{x \to 0} = \lim_{x \to 0} |xb| \to 0 \therefore \text{Well Conditioned} $$

As $x$ approaches infinity.. the condition number also approaches infinity.

$$ \lim_{x \to \infty} = \lim_{x \to \infty} |xb| \to \infty \therefore \text{Ill Conditioned} $$

The third case (i.e., $x \to -\infty$) is simliar to $x \to \infty$.