Derivative-free optimization

Have you ever tried to solve a puzzle but you couldn't figure out the answer even after trying several ways? That's where a concept called "derivative-free optimization" comes in.

Imagine that the puzzle is actually a mathematical equation that you need to solve, but you don't know how to take the "derivative" of the equation. A derivative is like a shortcut that allows you to find the rate at which something changes over time. However, in some cases, it can be really hard or even impossible to find the derivative of an equation.

Now, since you can't use the derivative for solving the puzzle or the equation, what can you do? That's where a method called "derivative-free optimization" comes in. This method uses a trial and error approach to find the solution to the equation.

Let's say you want to find the maximum value of the equation y=x^2+4x-5. To do that, you can start by guessing different values of x and plugging them into the equation to see what y value you get. If you keep trying different values of x and see that as you increase x, y starts to increase too, then you are on the right track. You can continue by trying values of x that are closer to the maximum value, until you have found the maximum value of y.

So, derivative-free optimization is a method of finding the best solution to a problem based on trial and error, without using the shortcut of the derivative. It's a bit like trying different solutions to a puzzle until you find the one that works best.