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

Derivatives

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.