Section 2.1 Derivative Function

Definition of the Derivative Function
The derivative of f(x) with respect to x is the function f'(x) such that

f'(x)= lim_{(h – >0)} \(\frac{f(x+h) - f(a)}{h}\)

If this limit exists
A function f(x) is differentiable at a point (a, f(a)) if f'(x) exists

\(\frac{dy}{dx}\)= f'(x)= y’

Section 2.2 The derivatives of Polynomial functions

The constant function rule

f(x)=c => f'(x)=0   or  \(\frac{dc}{dx}\)=0

The power rule for y= xⁿ is

\(\frac{d}{dx}\) (xⁿ) = nxⁿ⁻¹

The constant multiple rule for f(x)=c*h(x) is

f'(x)=c*h'(x)

The sum rule for f(x)= g(x) + h(x) is

f'(x)= g'(x) + h'(x)

The difference rule for f(x)= g(x) – h(x) is

f'(x)= g'(x) – h'(x)

Section 2.3 The Product rule

If f(x)=u(x) × v(x) then the product rule is
f'(x)='(x) × v(x) + u(x) × v'(x)
The power of a function rule for integers for f(x)=(u(x))ⁿ is
f'(x)=n × (u(x))ⁿ⁻¹  × u'(x)

Section 2.4 The Quotient rule

If f(x)=\(\frac{u(x)}{v(x)}\)  , then

f'(x)=\(\frac{ u'(x)v(x) - u(x)v(x)}{v²(x)}\)

Section 2.5 The Derivatives of composite functions

Definition: A composite function of u(x) and v(x) is

u⁰v(x) = u(v(x))

The Chain rule

If 2 function u(x) and v(x) are continuous and derivable, then

f(x)=u(v(x)) => f'(x)=u'(v(x)) × v'(x)