Scalar Derivative and Partial Derivatives

Derivative

In calculus, derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). The process of finding a derivative is called "differentiation"

'The derivative of' is commonly written as $\frac{d}{dx}$ or $f'(x)$

Find a Derivative

Example:

$f'(x)=\frac{d}{dx}x^2 = 2x$

means the slope or "rate of change" at any point in the graph/function $x^2$ is 2x. So when x=2 the slope is 2x = 4

Scalar derivative rules

Partial Derivatives

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

'The partial derivative with respect to x' is written as $\frac{\partial f}{\partial x}$ or $f'_x$where ${\partial}$is called 'del' or 'curly dee'.

Find Partial Derivatives

Consider a function with two variables (x and y): $f(x,y) = x^2 + y^3$

a) Partial derivative with respect to x (Treat y as a constant like a random number 12)

$$f'_x = \frac{\partial f}{\partial x} = 2x+0=2x$$

b) Partial derivative with respect to y (Treat x as a constant)

$$f'_y =\frac{\partial f}{\partial y}= 0+3y^2=3y^2$$

Link: - Matrix Calculus For Deep Learning - MathIsFun: Introduction to Derivatives - MathIsFun: Partial Derivatives - Khan Academy: Basic Derivative Rules