What is the derivative of a log?

Cracking the Code: Derivatives of Logarithmic Functions (Explained!)

Logarithmic functions. They might sound intimidating, but they’re actually super useful for modeling all sorts of things, from population growth to how quickly your coffee cools down. And if you’re diving into calculus, knowing how to take their derivative is absolutely key. So, let’s break it down, shall we?

First, a quick refresher. What is a logarithm, anyway? Simply put, it’s the answer to the question: “What power do I need to raise a certain number (the base) to, in order to get another number?”. Think of it like this: log₁₀(100) = 2, because 10² = 100. Easy peasy.

Now, when it comes to calculus, there are two main types of logs you’ll run into:

• The Natural Logarithm (ln): This is the cool kid, the one everyone uses. Its base is e (Euler’s number, roughly 2.71828). So, ln(x) basically asks, “What power do I need to raise e to, to get x?”.
• The Common Logarithm (log): This one’s base 10. If you just see “log(x)”, without a base specified, you can usually assume it’s base 10.

Okay, enough review. Let’s get to the good stuff: derivatives!

The Magic Formula for the Natural Logarithm’s Derivative

Here’s the thing: the derivative of ln(x) is surprisingly simple. Ready for it?

If f(x) = ln(x), then f'(x) = 1/x.

That’s it! Seriously. It means that the instantaneous rate of change of ln(x) is just the reciprocal of x. Pretty neat, huh?

Want to know why this works? Buckle up for a tiny bit of math. We can prove it using something called the “first principle of differentiation”. It’s a bit technical, but trust me, it’s worth seeing once. It involves limits and some logarithmic properties, and after a few steps, you’ll arrive at the 1/x result. I won’t bore you with the full derivation here, but if you are interested, you can find it in the previous version of this article.

What About Logs With Other Bases?

So, what if you’re dealing with a log that isn’t a natural log? No problem! There’s a formula for that too. If f(x) = logₐ(x) (where a is any base), then:

f'(x) = 1 / (x * ln(a))

Notice something cool: if you plug in e for a, you get back the derivative of the natural log, since ln(e) = 1. Everything’s connected!

Again, there’s a proof for this, and it involves a technique called “implicit differentiation.” The basic idea is to rewrite y = logₐ(x) as aʸ = x, and then differentiate both sides. It’s a clever trick that gives you the formula we just talked about.

The Chain Rule: Your New Best Friend

Now, things get really interesting when you combine logs with other functions. This is where the chain rule comes in. Remember the chain rule? It’s how you differentiate composite functions (functions inside of other functions).

Let’s say you have f(x) = ln(g(x)), where g(x) is some other function. Then, the chain rule tells us:

f'(x) = g'(x) / g(x)

Basically, you take the derivative of the “inside” function (g'(x)), and divide it by the original “inside” function (g(x)).

For a general logarithm f(x) = logₐ(g(x)), you get:

f'(x) = g'(x) / (g(x) * ln(a))

Example Time!

Let’s find the derivative of ln(x² + 4).

Here, g(x) = x² + 4, so g'(x) = 2x. Using the chain rule, we get:

f'(x) = (2x) / (x² + 4)

See? Not so scary after all!

Why Bother With Logarithmic Derivatives?

Okay, so you know how to find these derivatives, but why should you care? Well, they pop up all over the place!

• Optimization: Need to find the maximum or minimum value of a function involving logs? Derivatives are your go-to tool.
• Related Rates: Tracking how quickly things are changing when logs are involved? Derivatives to the rescue!
• Logarithmic Differentiation: This is a super-powerful technique for differentiating really complicated functions. The trick? Take the logarithm of both sides first, which often simplifies things dramatically. I’ve used this countless times to make my life easier!
• Simplifying Complex Functions: Sometimes, just using logarithmic properties can make a function easier to differentiate in the first place.

Final Thoughts

The derivative of a log might seem like a small piece of the calculus puzzle, but it’s a surprisingly important one. Master these formulas, get comfortable with the chain rule, and you’ll be well-equipped to tackle a wide range of problems. So go forth, differentiate, and conquer!