Derivative

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 $\frac{d(x)}{dx}$ for a function $f(x)$ is literally the slope formula. A tiny change in the rise (aka the $y$) over the tiny change in run (aka the $dx$).

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:

• Rate of Change
• Implicit Differentiation
• Concavity

Partial Derivatives:

Uni never actually taught me this…

Imagine you have $f(x,y)$. The partial derivative ($\frac{\partial f}{\partial x}$) tells you how much the output of $f$ changes with respect to $x$, assuming $y$ 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.