Wouldn't it be much better if you could cross out dy/dx to become y/x? Unfortunately, that's not the case since our good friend Gottfried Leibniz decided that wasn't possible. Though, to be fair it's not his fault.
A derivative is an expression representing the rate of change of a function with respect to an independent variable. Basically, it allows us to calculate the gradient—or steepness—of a line that touches a curve at a certain point. What does it mean for a line to "touch" a curve? Take a look at this image.
Notice how the green line is parallel with the red curve at point A. From this information, we can conclude that the green line has the same steepness as the curve at point A. This steepness is measured with the derivative of either function at point A.
Let's say a variable h exists that has a value of 3.
Point B is 3 units to the right of point A. If we draw a line from point A to B, it has a certain gradient. This is not yet the exact steepness of the curve at point A since 3 is still too big. Let's try shrinking h down to 0.1.
Now that points A and B are much closer, we're getting closer to our desired steepness. If we continue with this process, shrinking h down and down, we will eventually get the exact gradient of the curve at point A. This is how a derivative is defined. So, a derivative is the gradient of a curve, measured with two points that are close together: Point A whose coordinates are (x, f(x)) and point B whose coordinates are (x+h, f(x+h)). As the value of h approaches zero, our gradient will become more and more accurate until eventually, we get the steepness of the function at point A.
We know that the gradient of a function is calculated by (y₂ - y₁) / (x₂ - x₁), so if we substitute the coordinates of points A and B into this function, we get the formal definition of a derivative:
I hope you guys enjoyed this brief introduction to derivatives. If you've got any questions, feel free to ask in the comments! I'll try to answer every question. Also, I'm open to feedback!