Lecture 5 Difference Quotients and Derivatives

f ‘ (a) = slope of tangent at (a, f(a)) Should be “best approximating line to the graph at the point (a,f(a))” Need not exist

Tangent Line at (a,f(a)) Approximating Secant Line

Idea of Definition of f ‘ (a) If tangent line, T, exists the secant lines are better and better approximation If secant lines approximate T then their slopes should approximate the slope of T In limit slopes of secant lines should become slope of T

Definition of f ‘ If f ‘ (a) exists then the tangent line to the graph of f(x) at x =a is the line through (a,f(a)) with slope f ‘ (a). Thus if f ‘ (a) exists the equation of the tangent line to the graph of f(x) at (a,f(a)) is If f(x) is a function and a is in the domain of f then if it exists

The Derivative Function If f(x) is a function then there is a new function given by the correspondence x -> f ‘ (x) This new function is called the derived function of f or the derivative of f and is simply denoted by f ‘ (x) Variable need not be “x”. If f is a function of t the f ‘ is a function of t also

Eqn of Tangent Line at x = 3 f ‘ (a) = f ‘ (3) is the slope of this tangent line or approximately -3/2 (from graph) Here: a = 3, f(a) = -3 (from graph) Point (a, f(a)) = (3, -3) Equation of tangent line is y –(-3)) = (-3/2)(x -3) May be put in standard form such as y = 3/2 – (3/2)x.

Differentiable Function The function f is said to be differentiable on the interval (a,b) if f ‘ (x) exists for every x in (a,b). e.g. if f(x) = |x| then f is differentiable on the intervals (-1,0) and (0,1) but not differentiable on (-1,1)

Why Differentiable Is So Useful If f is differentiable at x=a then for the purpose of solving problems which only involve the behavior of f “near a”, f can be replaced by the function whose graph is the tangent line. T(x) is the function whose graph is the tangent to f(x) at x = 0

Differentiable Implies Continuous If the graph of f(x) is “broken” at x = a then f cannot have a tangent line at x=a. A function whose graph has a gap at x =a cannot be replaced “near a” by one whose graph is a line. Thus if we can compute f ‘ (a) using the derivative rules (even in principle) then We know that f is continuous at x =a This function is not continuous at x =1 since the graph is broken at x =1. This means that it is not differentiable at x =1

It is possible to be continuous but not differentiable

Calculation of f ‘ (x) for some simple functions f. f(x) = c, c a constant f ‘ (x) = = f (x) = x =

Derivative of

Calculating Tangent Lines Analytically The equation of the tangent line to the graph of f(x) at the point (a, f(a)) is Y = f(a) + f ‘ (a) ( x –a) From general point-slope eqn for a line Here

Derivative Rules If f is a function and c a constant then (cf) ‘ = c (f ‘) If f and g are functions then (f+g) ‘ = f ‘ + g ‘ (Leibnitz Rule) If f and g are functions then (fg)’ = f’ g + f g’ Each of the above works when the derivatives on the right exist. It is possible for the left side derivatives to exist when the ones on the right do not

Rules break hard problems into one or more simpler problems

f ‘ (x) = ( )’ ( ) + ( ) ( )’ Another example f ‘ (x) = (3 – 2 x ) ( 14 x+1) + ( ) ( 14)

Derivative of f(x) =

Tangent Line to at x = -.8 y – f(a) = f ‘ (a) (x –a) a = -.8 f(a) =.64 Since f ‘ (x) = 2x, f ‘ (a) = 2a = 2(-.8) = -1.6 So equation of the tangent line is y –.64 = (-1.6)) (x –(-.8) ) or y = x

Chain Rule Power rule is a special case where

More Derivative Rules ( ) ‘ = ( )‘ ‘ Quotient Rule Power Rule

There are really only 3 Rules – ( f+g ) ‘ = f ’ + g ‘ –The product rule –The chain rule The other rules come from these For example consider the rule (cf)’ = c f ‘ if c is a constant By the product rule But we know c ‘ = 0 so (c f ) ‘ = c f ‘

Quotient Rule Comes from the Product Rule