Converting exponentials to base e

As mentioned in the last topic, the gradient of is exactly the same as the value at that point.

That makes a lot of things much easier for us to work with!

We can actually use this power of base e exponentials with any exponential function - by converting them to base .

The key to converting any exponential function to base is the fact that we can rewrite any number as .

Basically, what we’re doing here is finding what number we need to raise to in order to get - and that number is .

Then, we raise to that power to get back.

While it might seem like we’ve complicated things, we now have a term in base , which we can find the gradient of!

Converting to base

Knowing that , we can rewrite as .

Then, we can use the rule of indices that says that to rewrite as .

Now we have rewritten as , we can find the gradient of it using the rule that (see base e exponentials for more on that rule).

In this case, , so (because is a constant, so it just comes out of the differentiation).

Then, we can apply the rule to get .

Finally, we can rewrite back to to get the final answer of .

for any and .