Fundamentals of derivatives

This post is a quick recap on some fundamentals of calculating derivatives, useful for image processing and deep learning.


Calculating the derivative

Given x to the power of n, where n ≠ 0.

$$ f(x) = x^n $$ $$ f'(x) = nx^{n-1} $$

This is known as the power rule. For example:

$$ \frac{d}{dx}(3x^7+4x^2-x+100) = 21x^6+8x-1 $$

Derivative of a sum of functions

The derivative of a sum of functions is the sum of the derivatives:

$$ f(x) = u(x) + v(x) $$ $$ f'(x) = u'(x) + v'(x) $$

So simply compute the derivatives one at a time then sum them individually.

Derivative of a product of functions

The derivative of a product of functions is defined:

$$ f(x) = u(x) \cdot v(x) $$ $$ f'(x)=u'(x)\cdot v(x)+u(x)\cdot v'(x) $$

This is known as the product rule.

Derivative of composed functions

For function composition the derivative is:

$$ f(x) = u(v(x)) $$ $$ f'(x) = u'(v(x)) \cdot v'(x). $$

This is known as the chain rule.

If stuck, play with this calculator.

About this blog

Hello, this is my first blog post with this new website made with Jekyll.

All the content is written in a simple markdown that supports HTML and CSS, and it is hosted for free on Github Pages.

The nice thing about this in comparison to traditional Wordpress or Tumblr is that I can quickly edit files locally (e.g. inside of vim or Atom), and push commits which are reflected instantly.