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Lecture

# TheDerivative.pdf

23 Pages
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Department
Mathematics
Course
MATA30H3
Professor
Sophie Chrysostomou
Semester
Winter

Description
THE DERIVATIVE 3.1 DERIVATIVES Problem: Find the slope of the tangent line to the curve of f(x) = x at 2 the point (1,1) and at the point (a,a ). 68 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES The Tangent Line Problem (revisited) We now generalize how we ﬁnd the equation of the tangent line to the curve of the function y = f(x), at the point P(a,f(a)). For h ▯= 0 let Q(a+h,f(a+h)) be on the graph and get the slope of the secant PQ: l . . m = PQ . . 0 Rates of Change Let y = f(x) be a function. If f(x ) = 1 , and1f(x ) = y ,2the ch2nge in x, called the increment of x is ∆x = x −x and2the c1rresponding change in y, called the increment of y is ∆y = y − y = f2x ) 1 f(x ). 2 1 l . . . . The slope of the secant PQ is also called the average rate of change of y ∆y f(x 2 − f(x ) 1 with respect to x and is: = . ∆x x2− x 1 The instantaneous rate of change of y with respect to x is: ∆y f(x 2 − f(x )1 ∆x→0 = x2→x 1 . (2) ∆x x 2 x 1 Substituting a for x a1d a + h for x we o2tain the limit of type (1). 69 ▯2013 by Sophie Chrysostomou THE DERIVATIVE DERIVATIVE OF f Limits of the form (1) & (2) arise often and, they are given a special name and notation. DEFINITION: Let y = f(x) be a function deﬁned on an open interval (c,d) and let a ∈ (c, d). • We say f is diﬀerentiable at x = a if the limit def f(x) − f(a) f(a + h) − f(a) f (a) = lim = lim x→a x − a h→0 h exists. • This limit above is called the derivative of f at a and is also denoted by ▯ ▯ ▯ dy ▯ df(x) ▯ d ▯ ▯ , ▯ , f(x) ▯ , D fxx)| x=a dx x=a dx x=a dx x=a • The derivative of f is the function deﬁned by: f (x) = lim f(x + h) − f(x) , or f (x) = lim f(x) − f(x ) 1 h→0 h x1→x x − x 1 for all x for which the limit on the right hand side of the equation above exists. • f is diﬀerentiable on an interval (c, d) [or (c, ∞) or (−∞, d) or (−∞, ∞)], ′ if f (x) exists for all x in the interval. (Note that all these intervals are open!) ′ • The tangent line to y = f(x) at the point (a,f(a) has slope f (a) and equation: ′ y − f(a) = f (a)(x − a). 70 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES x EXAMPLE: Find the tangent line at a = 1, of the graph of f(x) = . x + 3 HOMEWORK: (a) Find the slope of the tangent lines to the graph of f(x) = x at the points (0,0),(1,1),(3,9),(−1,1),(−3,9). 71 ▯2013 by Sophie Chrysostomou THE DERIVATIVE √1 ′ EXAMPLE: Let f(x) = x. Find f , the derivative of f, and its domain. 72 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES EXAMPLE: Where is f(x) = |x − 1| diﬀerentiable? 73 ▯2013 by Sophie Chrysostomou THE DERIVATIVE Points where a function is not diﬀerentiable. Case 1. “Corner” Case 2. Discontinuity Case 3. Vertical Tangent 74 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES Theorem : If f is diﬀerentiable at a, then f is continuous at a. (Note: The converse is false. Functions can be continuous but not diﬀeren- tiable at a point a.) ′ Higher Derivatives: If f is diﬀerentiable, then its derivative f is a function. If f is diﬀerentiable then its derivative is denoted by f and is the second derivative of f. Successively we may get the the third, fourth ..., derivatives of f. More notations for higher derivatives are: 2 3 2 d y = d (dy ), d f = d (dy ), f (30(x)▯▯▯ and so on. dx2 dx dx dx3 dx dx 75 ▯2013 by Sophie Chrysostomou THE DERIVATIVE 3.2 RULES FOR DIFFERENTIATION 1. Derivative of a constant function: If f(x) = c for some constant c, then f (x) = 0. Proof: 2. Power Rule: If f(x) = x , then f (x) = nx n−1 for any nonzero integer n. Proof: For positive integers only. 76 ▯2013 by Sophie Chrysostomou 3.2 RULES FOR DIFFERENTIATION 3. Linear Combination Rule: If f and g are diﬀerentiable at x ∈ R and if r,s ∈ R are constants, then rf + sg is diﬀerentiable at x and ′ ′ ′ (rf + sg) (x) = rf (x) + sg (x) At this point using rules 1-3 we can ﬁnd the derivative of any polynomial: 4. Product Rule: If f and g are diﬀerentiable at x ∈ R then f ▯ g is diﬀer- entiable at x and (f ▯ g) (x) = f (x)g(x) + f(x)g (x) ′ 77 ▯2013 by Sophie Chrysostomou THE DERIVATIVE f(x) 5. Quotient Rule: If h(x) = g(x) and f and g are diﬀerentiable at x and g(x) ▯= 0 then h is diﬀerentiable at x and: ′ ′ h (x) = f (x)g(x) − g (x)f(x) . (g(x)) 2 The proof is left as homework! 6. Chain Rule: If g is diﬀerentiable at x, and f is diﬀerentiable at g(x), then f ◦ g is diﬀerentiable at x and (f ◦ g) (x) = f (g(x)) g (x) on in Leibniz notation for y(u(x)): dy dy du = ▯ dx du dx 2 100 EXAMPLE: Let r(x) = ( x ) . Find r (x) for x ▯= 1. (x−1)
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