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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?
- a function
- * provided the limit exists
when is function f(x) called differentiable?
- at x=a if limit f'(a) exists
- differentiable everywhere 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) = x2 + 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) = xn
- f'(x) = nxn-1
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'
- if u(x) and v(x) are differentiable then
what should you always do before trying to solve for a derivative?