Definition
Often referred to as “The Instantaneous Rate of Change”. Alternatively, consider that that’s also the definition of a slope of a function. Thus, the derivative also gets you the slope of a curve. Since TIROC is quite oxymoronic (explained below), it’s more mathematically accurate to refer to it as “The Best (constant?) Approx for a Rate of Change”.
Oxymoronic:
Consider the definition The Instantaneous Rate of Change. That’s a Paradox in and of itself. How can something be instant (at a single point in time) and change (a difference between two points in time)?
Thinking of it as a slope just makes sense. The term for a function is literally the slope formula. A tiny change in the rise (aka the ) over the tiny change in run (aka the ).
To go from a function to it’s derivatives, there’s a certain art to it. Some really smart people discovered the art and patterns and made some rules. It’s useful to know the art behind it as well.
Related:
Partial Derivatives:
Uni never actually taught me this…
Imagine you have . The partial derivative () tells you how much the output of changes with respect to , assuming does NOT change.
Mathematically, it’s the same as a regular derivative but applied to one variable at a time. As well, you’re only focusing on deriving any terms with the variable of choice, dropping everything else.
Gradient:
The gradient is a vector that points in the direction of the steepest increase of that function:
- The magnitude tell you how steep it is (instantaneously).
- If gradient is 0, you’re at a local minimum, maximum or saddle point.
- You can get the gradient of a function by partially deriving with respect to every variable, and then putting them in a vector.
- Boom - it’s also useful in Gradient Descent.