The third partial derivative over ** x** is:

Deriving it is left as an exercise (but see Appendix1 in previous post).

]]>It’s said that communication is 55% visual (expression, posture, etc), 38% vocal (tone, inflection, etc), and only 7% verbal (or textual). So, most forms of online communication (like comments) are severely stunted in bandwidth. Writing is so much different from speaking! We all have a lifetime’s practice speaking, but not everyone has training and practice writing.

Engineering, despite its reputation, is easy and relaxing compared to people. Not easy. Not relaxing. Most of them, anyway. The thing about math is that opinions don’t matter. Math is right or not. Generally the case with science or engineering, too. Electricity doesn’t care what you think.

I’d guess psych-people, vested in helping people cope, focus on what is, and right enough, but I am ever nagged (and perhaps you are, too) about *ought*. My Weltschmerz sometimes reaches profound levels.

Anyone of those three actors can go awry at any time for any number of reasons. Without intention.

For an Engineer, it must be super frustrating.

My psych-girl suggested to me once that I “lower my expectations”. That’s hard for me.

Hang in there. Have a good weekend. cheers.

]]>I… really, really, really hate WP software. It seems to be designed, built, and maintained, by idiots. (Back in the last century, software was perceived as the booming industry and a lot of people got into it who had no instinct or talent for it, which is why so much of it is so bad.) I just spent 20 minutes writing a reply and lost it because I accidentally hit a wrong key and this incompetent WP software can’t trap the exit and ask if I’m sure I want to just throw away 20 minutes work. Taking care with a user’s work is software development 101, and these fucks failed.)

Speaking of which, were you looking at my post on my blog’s website (logosconcarne URL) or with the WordPress Reader (wordpress URL)? It’s only the latter that’s the problem.

The comment I lost, I was off on a long-standing rant about the war between the original vision Tim Berners-Lee had for the web versus what it became. Not the fraud or commercialization or porn or social media, but just in how content is rendered. The original idea was that end users had a lot of control over how pages look. Artists and content developers wanted their stuff to look exactly as designed. Over time, they won, and the web became very complicated to allow artistic control in a medium that wasn’t originally designed for it.

This is like a bad version of that. The WP Reader wants all posts to look identical, so it throws away a lot of the meta-information writers include (sometimes even bolding and italics get lost). For instance, my section marks (§) are supposed to be centered, but the WP Reader has them against the left margin. And now it’s enlarging most (but weirdly not all) images so they stretch from left margin to right margin (in the Reader). That blows them up and they look bad.

[sigh] WordPress… just one more thorn in my side. I just finished a long round of emails with their support staff over an issue I had with the editor. Used to be interacting with WP Help always had a good result. That hasn’t been true in years now. It always ends in frustration and no fix.

]]>Each device (phone, tablet, laptop) I have presents the content differently.

It’s frustrating, yes. As well as the constant “upgrades “.

A constant race to be “better “ leads to wtf ! Why the hell did they do that?

Analytics?

]]>It’s worth digging into the mechanics of taking the derivative of the exponential function. It begins with the basic fact that the derivative of the exponential function ** is** the exponential function:

The slope of ** e^{x}** at

The chain rule is:

If a function, **f**, depends on a sub-function, **g**, multiply the derivative of the outer function times the derivative of the inner function.

To see how this works, we can see what happens when we apply the chain rule to the basic exponential function (which we know derives to itself). We’ll treat the plain ** x** in the exponent as a function,

Applying the chain rule and setting ** u=g(x)** to simplify the exponential:

Since the derivative of ** x** is just

Where ** u** is some expression containing (at least one occurrence of)

With that understanding, let’s try this:

Because the derivative of **2 x** is just

Here’s one with a *square* of ** x**, such as appears in the Gaussian exponential function:

Because the derivative of ** x²** is

If there was a constant in front of the exponential, it just gets multiplied by the derived exponent:

We can break this down as an instance of the product rule, which is:

But since ** a** is a constant, it derives to zero, so:

Which gives us the result shown above.

Finally, note that, since a constant derives to zero, an exponential function with a constant exponent also derives to zero:

Hopefully this helps make it clear how to derive more involved exponential functions!

]]>This appendix shows step-by-step how to calculate the derivatives used in the post.

Here’s the base equation:

Where ** a** controls the amplitude and

Then along the **Y** axis:

The derivative uses the chain rule where we equate ** u** to the exponent in the exponential:

So, with ** u** a function of

Because the derivative of an exponential is that exponential. Now we only need to derive:

With a similar derivation for ** y**. Plugging these into the above, for

Which, rearranged, is what’s shown above. The * y* derivation works the same way.

Deriving the base equation gives almost the same result, we just include the constant, ** a**:

With a similar one for ** y**.

**§**

For the second derivative we need the product rule:

Which is why we derived just the exponential for later. This is later. In the context of the product rule, we need to derive:

And we already know what ** v‘(x)** is, so we just need

Putting it all together, for ** x**:

Extracting the common exponential, doing the multiplication on the left, and multiplying the right term by c²/c² to create a common denominator:

Cleaning it up we have the second derivation:

With a similar one for ** y**.