MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10 Related Rates 7.1 One-to-one functions. Inverse Deadline to notify us if you have a conflict with Test 1: June 05
Derivative Definition (Derivative) Let f be a function and a be a number in the domain of f. We say that f is differentiable at a if f (x) f (a) lim x a x a exists. If this limit exists, it is called the derivative of f at a and is denoted by f (a). If we let h = x a, then x a iff h 0. Therefore f f (x) f (a) f (a + h) f (a) (a) = lim = lim. x a x a h 0 h
The Derivative as a Function We can view the derivative as a function, with f f (x + h) f (x) f (y) f (x) (x) = lim = lim. h 0 h y x y x We say f is the derivative of f. The domain of f is the set {x : f (x) exists}.
Example 4 Example Find f (0) where f (x) = { sin x if x 0 x if x > 0. Recall Definition if the limit exists. f f (x + h) f (x) f (y) f (x) (x) = lim = lim. h 0 h y x y x
Example 5 Example For f (x) = x, find f. State the domain of f.
Example 5 Example For f (x) = x, find f. State the domain of f. y y f (x) = x x f (x) = 1 2 x x
Example 6 Example Show that the function f (x) = x is not differentiable at 0. y x
Example 6 Example Show that the function f (x) = x 1 3 is not differentiable at x = 0. y x
Differentiability Continuity Theorem If f is differentiable at a, then f is continuous at a. WARNING: The converse is NOT true.
Differentiability Continuity Theorem If f is differentiable at a, then f is continuous at a. WARNING: The converse is NOT true. y x f (x) = x is continuous at x = 0, but NOT differentiable at x = 0.
Differentiability Continuity Theorem If f is differentiable at a, then f is continuous at a. WARNING: The converse is NOT true. y Corollary x f (x) = x is continuous at x = 0, but NOT differentiable at x = 0. If f is not continuous at a, then f is not differentiable at a.
Failure of Differentiability There are three ways in which a function can fail to be differentiable. y y y x x x (a) Corner or Cusp (b) Vertical tangent (c) Discontinuity
Some Differentiation Formulas Proposition Proof. Exercise. IF f (x) = α, α R, THEN f (x) = 0 for all x. IF f (x) = x, THEN f (x) = 1 for all x.
Some Differentiation Formulas Theorem (Derivatives of Sums and Scalar Multiples) Let α R. IF f and g are differentiable at x, THEN f + g and αf are differentiable at x. Moreover, 1 (f + g) (x) = f (x) + g (x), derivative of sum is sum of the derivatives 2 (αf ) (x) = αf (x). derivative of scalar multiple is scalar multiple of derivative
Some Differentiation Formulas Theorem (The Product Rule) If f and g are differentiable at x, then so is their product and (f g) (x) = f (x)g (x) + g(x)f (x).
Some Differentiation Formulas Theorem (The Product Rule) If f and g are differentiable at x, then so is their product and (f g) (x) = f (x)g (x) + g(x)f (x). Exercise Prove that for each positive integer n, IF p(x) = x n, THEN p (x) = nx n 1. Hint: Use Mathematical Induction.
Some Differentiation Formulas Theorem (The Product Rule) If f and g are differentiable at x, then so is their product and (f g) (x) = f (x)g (x) + g(x)f (x). Exercise Prove that for each positive integer n, IF p(x) = x n, THEN p (x) = nx n 1. Hint: Use Mathematical Induction. Proposition (Derivative of Polynomials) IF P(x) = a n x n + a n 1 x n 1 + + a 2 x 2 + a 1 x + a 0, THEN P (x) = na n x n 1 + (n 1)a n 1 x n 2 + + 2a 2 x + a 1.
Some Differentiation Formulas Example Suppose that g is differentiable at each x R and that f (x) = (x 3 2x)g(x). Find f (x) given that g(2) = 3 and g (2) = 1. Theorem (The Product Rule) If f and g are differentiable at x, then (f g) (x) = f (x)g (x) + g(x)f (x).
Some Differentiation Formulas Theorem (The Quotient Rule) If f and g are differentiable at x and g(x) 0, then the quotient f g is differentiable at x and ( ) f (x) = g(x)f (x) f (x)g (x) g [g(x)] 2.
Some Differentiation Formulas Theorem (The Quotient Rule) If f and g are differentiable at x and g(x) 0, then the quotient f g is differentiable at x and ( ) f (x) = g(x)f (x) f (x)g (x) g [g(x)] 2. Exercise Show that for each integer n, Even more, for any c R, IF p(x) = x n, THEN p (x) = nx n 1. IF p(x) = x c, THEN p (x) = cx c 1.
Some Differentiation Formulas Example Let f be a differentiable function, and let g(x) = f (x) x. Find g (x). Theorem (The Quotient Rule) If f and g are differentiable at x and g(x) 0, then ( ) f (x) = g(x)f (x) f (x)g (x) g [g(x)] 2.
The d dx Notation Leibniz Notation If y = f (x), we write f (x) = dy dx Example If y = x 3, then
The d dx Notation Leibniz Notation If y = f (x), we write f (x) = dy dx Example If y = x 3, then dy dx = 3x 2.
The d dx Notation Leibniz Notation If y = f (x), we write Example f (x) = dy dx If y = x 3, then dy dx = 3x 2. d The symbol can also be used as prefix before the expression to be dx differentiated. Example d dx (x 2 + 2x) = 2x + 2
The d dx Notation The notation dy dx is used to emphasize the fact that we are evaluating x=a the derivative dy at a. dx Example If y = 3x 1 dy. Find 5x + 2 dx. x=0
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) =(f ) (x) = 20x 3
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) =(f ) (x) = 20x 3 f (x) =(f ) (x) = 60x 2
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) =(f ) (x) = 20x 3 f (x) =(f ) (x) = 60x 2 f (4)(x) =(f ) (x) = 120x.
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) = (f ) (x) = 20x 3 f (x) = (f ) (x) = 60x 2
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) = (f ) (x) = 20x 3 f (x) = (f ) (x) = 60x 2 f (4) (x) = (f ) (x) = 120x.
Derivatives of Higher Order Let f (x) = x 5, then f (x) = 5x 4 f (x) = (f ) (x) = 20x 3 f (x) = (f ) (x) = 60x 2 f (4) (x) = (f ) (x) = 120x. Notation f (n) denotes the n th defivative of f. Example Consider a polynomial of degree n, P(x) = a n x n + a n 1 x n 1 + + a 2 x 2 + a 1 x + a 0 What is the k th -derivative, P (k), for k > n?
Derivatives of Higher Order In Leibniz notation the derivative of higher order are written d 2 y dx 2 = d dx ( ) dy, dx d 3 y dx 3 = d dx ( d 2 ) y dx 2,...
Derivatives of Higher Order Example Let f (x) = 1 x. 1 Find f (x). 2 Find a general formula for the n th -derivative, f (n).
Derivative of Trigonometric Functions Theorem The function sine is everywhere differentiable and x is measured in radians Recall that Definition if the limit exists. d sin x = cos x dx f (x) = lim h 0 f (x + h) f (x) h
Derivative of Trigonometric Functions Theorem The function cosine is everywhere differentiable and Proof. Exercise. Hint: Use the following identity d cos x = sin x dx cos(x ± y) = cos x cos y sin x sin y
Example 1 Derivative of Trigonometric Functions Find the derivative of f (x) = tan x.
Derivative of Trigonometric Functions d d sin x = cos x dx d d cos x = sin xx dx d dx tan x = sec2 x csc x = csc x cot x dx sec x = sec x tan x dx d dx cot x = csc2 x Exercise Prove these formulas.
Example 2 Derivative of Trigonometric Functions Find the 27 th derivative of f (x) = sin x.
The Chain Rule Example 1 Let F (x) = (2x + 5) 3. Find F (x) (a) by expanding (2x + 5) 3 and differentiating the resulting polynomial. (b) using the product rule.
The Chain Rule Example 1 Let F (x) = (2x + 5) 3. Find F (x) (a) by expanding (2x + 5) 3 and differentiating the resulting polynomial. (b) using the product rule. 2 Now, compute the derivative of G(x) = (2x + 5) 2016
The Chain Rule Example 1 Let F (x) = (2x + 5) 3. Find F (x) (a) by expanding (2x + 5) 3 and differentiating the resulting polynomial. (b) using the product rule. 2 Now, compute the derivative of We need a better approach. G(x) = (2x + 5) 2016
The Chain Rule Theorem (Chain Rule) If g is differentiable at x and f is differentiable at g(x), then the composite function f g defined by (f g)(x) = f (g(x)) is differentiable at x and (f g) (x) = f (g(x)) g (x).
The Chain Rule Theorem (Chain Rule) If g is differentiable at x and f is differentiable at g(x), then the composite function f g defined by (f g)(x) = f (g(x)) is differentiable at x and (f g) (x) = f (g(x)) g (x). In Leibniz notation, if y = f (u) and u = g(x), then where dy du is evaluated at u = g(x). dy dx = dy du du dx,
The Chain Rule Theorem (Chain Rule) (f g) (x) = f (g(x)) g (x). Example Let G(x) = (2x + 5) 2016. Find G (x). In general, Proposition If f is differentiable and n Z, then d dx [f (x)]n = n[f (x)] n 1 d dx f (x).
The Chain Rule d dx f }{{} outer function g(x) }{{} evaluated at inner function = f }{{} ( g(x) ) }{{} g (x) }{{} derivative of outer function evaluated at derivative of inner function inner function
The Chain Rule Example Let y = cos(x 2 ). Find dy dx. Theorem (Chain Rule) (f g) (x) = f (g(x)) g (x).
Repeated Use of the Chain Rule Example Find the derivative of h(x) = sin ( 2 + ) x 2 + 1. Theorem (Chain Rule) (f g) (x) = f (g(x)) g (x).
Rates of Change Example Consider the linear function y = mx + b. y As x changes from x 0 to x 1, y changes m times as much: x 1 x 0 y 1 y 0 y 1 y 0 = m(x 1 x 0 ). x
Rates of Change Example Consider the linear function y = mx + b. y As x changes from x 0 to x 1, y changes m times as much: x 1 x 0 y 1 y 0 y 1 y 0 = m(x 1 x 0 ). x In general, if f is differentiable and y = f (x). The slope dy dx = f (x) gives the rate of change of y with respect to x. y f x
Rates of Change Example The area A of a square is given by the formula A = x 2, where x is the length of a side. How fast does the area change with respect to the length when the diameter is 0.5cm, 1cm, 3cm?
Rates of Change Example The area A of a square is given by the formula A = x 2, where x is the length of a side. How fast does the area change with respect to the length when the diameter is 0.5cm, 1cm, 3cm? y A = x 2 x = 0.5 x = 1 x = 3 A = 0.25 A = 1 A = 9 da x=0.5 = 1 da x=1 = 2 x=3 = 6 da dx dx dx x
Rate of Change Example Suppose that we want to inflate a spherical balloon in such a way that the radius increases at a constant rate of 2cm per second. At what rate should we pump air into the balloon when the radius is 1cm? The volume of a sphere of radius r is V = 4 3 πr 3
Graphs Example 1 The function f has domain R, satisfies f (0) = 2, and is continuous. Below is the graph of its derivative f : 4 f 3 2 1 2 1 0 1 2 3 4 1 2 Sketch the graph of f.
Graphs Example 1 Solution 5 f 4 3 2 1 2 1 0 1 2 3 4 1