The derivative of a function f(x) (noted as f'(x)) is defined as:
An alternate definition is:
However, because of how tedious it is to use the definition of a derivative every time the derivative of a function is needed, there are shortcuts known as differentiation formulas which reduce the amount of work needed to differentiate functions. Here, we will prove that these shortcuts work.
Shortcut 1: if then
- (Take the natural log of both sides)
- (Differentiate both sides with respect to x)
- (solve for y', substitute x^n for y, simplify)
Shortcut 2: if , or then ...
Proofs of cases coming soon.