what is the derivative of f(x) defined as?
derivative of f(x) is
provided the limit exists
what are the two interpretations o derivative?
1. slope o tangent line o function at that point
2. instantaneous rate o change o function at the point
derivative is also what?
* provided the limit exists
when is function f(x) called differentiable?
at x=a if limit f'(a) exists
if f'(x) exists for all x
what is the process of taking a derivative called?
when is f(x) not differentiable?
if f'(a) or f'(x) D.N.E. at x=a or all x values
what is diff b/twn f'(a) and f'(x)
f'(a) = number
f'(x) = function
write prime notation, liebniz notation
how to find derivative of f(x) = x
+ 3 with limits?
sub it into equation, use lim approaching h
constant function rule
if f(x) = k
f'(x) = 0
includes values like pi, and other numbers
if f(x) = x
f'(x) = nx
constant multiple rule
if f(x) = kg(x), where k is set real numbers and g(x) differentiable
f'(x) = kg'(x)
proof for constant multiple rule
let u(x) and v(x) be differentiable
(uv)'=u'v + uv'
if u(x) and v(x) and h(x) be differentiable
(uvh)'=u'vh + uv'h + uvh'
power of a function rule
when factoring (3x
if u(x) and v(x) are differentiable then
what should you always do before trying to solve for a derivative?
calculus derivatives 1