Definition of Derivative

Definition : Derivative of f(x)  at x=a

f’(a)=  lim _(x->a)  \(\frac{f(x)-f(a)}{x-a}\) or

f’(a)=  lim _(h->a)  \(\frac{f(a+h)-f(a)}{h}\)

Derivative at a Point First Principles

1. Given a function f(x) and a point x=a, find f(a)
2. Calculate f(a+h)
3. Calculate f(a+h)- f(a)
4. Determine \(\frac{f(a+h)-f(a)}{h}\)
5. Find  f’(a)=  lim _(h->a) \(\frac{f(a+h)-f(a)}{h}\)

Derivative Function First Principles

f’(a)=  lim _(h->a) \(\frac{f(x+h)-f(x)}{h}\) h ≠ 0

Differentiability Reasoning

A differentiable function is continuous: If f(x) is differentiable at x = a, then f(x) is also continuous at x = a.

Derivative of Polynomial Functions

The general equation is

\(\frac{d(ax^n)}{dx}\) =\(nax^{n-1}\)

Example :   \(\frac{d(x^3+2x^2+3)}{dx}\) =\({3x^2+4x}\)

Power Rule, Constant Multiple Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, Chain Rule

Basic Derivative Rules

Constant Rule : \(\frac{d}{dx}\) (c)= 0

Constant Multiple Rule :\(\frac{d}{dx}\) (cf(x))= cf’(x)

Power  Rule :\(\frac{d}{dx}(x^n)\) =\( nx^{(n-1)}\)

Sum Rule : \(\frac{d}{dx}\) (u(x)+v(x))= u’(x)+v’(x)

Difference  Rule : \(\frac{d}{dx}\)(u(x)-v(x))= u’(x)-v’(x

Product Rule :\(\frac{d}{dx}\) (u(x)*v(x))= u’(x)*v(x)+u(x)*v’(x)

Quotient Rule : \(\frac{d}{dx}\)(\(\frac{u(x)}{v(x)}\)) = \(\frac{(u'(x)×v(x)-u(x)×v'(x))}{(v(x))^2}\)

Chain Rule :\(\frac{d}{dx}\) u(v(x)= u’(v(x)*v’(x)