Lecture 2. Derivative. 1 / 26

Basic Concepts Suppose we wish to nd the rate at which a given function f (x) is changing with respect to x when x = c. The simplest idea is to nd the average rate of change of the function y = f (x) as x varies from x 0 = c to x 1 = x + h for a given (small) h. This ratio is given by change in f (x) change in x = y x = f (x1) f (x0) x 1 x 0 f (x + c) f (x) = h The ratio can be interpreted as the slope of the secant line from the point P(c, f (c)) to the point Q(c + h, f (c + h) 2 / 26

Basic Concepts y y=f(x) f(c+h) y Q f(c) P x c c+h x 3 / 26

Basic Concepts Denition The expression f (x + h) f (x) h is called a dierence quotient for the function f (x). Denition The derivative of the function f (x) with respect to x is the function f (x) given by f (x) := lim h 0 f (x + h) f (x). h The process of computing the derivative is called dierentiation, and we say that f (x) is dierentiable at x = c if f (c) exists. Other symbols used for denoting the derivative y f (x) dy dx df dx (x) d dx f (x) x f (x) 4 / 26

Basic Concepts Property The slope of the tangent line to curve y = f (x) at the point (c, f (c)) is m tan = f (c) Property The rate of change of f (x) (so-called instantaneous rate of change) with respect to x when x = c is given by f (c). 5 / 26

Basic Concepts y y=f(x) f(c) P(c,f(c)) c x 6 / 26

Basic Concepts Theorem If the function f (x) is dierentiable at x = c, then it is also continuous at x = c 7 / 26

Techniques of Dierentiation Theorem (Constant Rule) For any constant c, d dx [c] = 0 That is, the derivative of a constant is zero. Theorem (The Power Rule) For any real number n R, d dx [x n ] = nx n 1 In words, to nd the derivative of x n, multiply the original exponent n then subtract 1 from the exponent of x. 8 / 26

Techniques of Dierentiation Theorem (The Constant Multiple Rule) If c is a constant and f (x) is dierentiable, then so is cf (x) and d dx [cf (x)] = c d [f (x)] dx That is, the derivative of a multiple is the multiple of the derivative. Theorem (The Sum Rule) If f (x) and g(x) are dierentiable, then so is the sum f (x) + g(x) and its derivative is given by d dx [f (x) + g(x)] = d dx [f (x)] + d [g(x)] dx That is, the derivative of a sum is the sum of the separate derivatives. 9 / 26

Interpretations Property (Relative and Percentage Rates of Change) The relative rate of change of a quantity Q(x) with respect to x is given by the ratio relative rate of change of Q(x) = Q (x) Q(x) The corresponding percentage rate of change of Q(x) with respect to x is percentage rate of change of Q(x) = Q (x) Q(x) 100% 10 / 26

Interpretations Example The gross domestic product (GDP) of a certain country was N(t) = t 2 + 5t + 106 billion dollars t years after 2000. 1 At what rate was the GDP changing with respect to time in 2010? 2 At what percentage rate was the GDP changing with respect to time in 2010? We have N (t) = 2t + 5 N (10) = 2 10 + 5 = 25 Therefore the rate of change of the GDP in 2010 was N (10) = 2 10 + 5 = 25 billion dollars per year. The percentage rate of change of the GDP in 2010 was per year. N (10) N(10) 25 100% = 100% 9, 77% 256 11 / 26

Interpretations Example If the position at time t of an object moving along a straight line is given by s(t), then the object has velocity v(t) := s (t) = ds dt and acceleration a(t) := v (t) = dv dt The object is advancing when v(t) > 0, retreating when v(t) < 0 and stationary when v(t) = 0. It is accelerating when a(t) > 0 and decelerating when a(t) < 0. 12 / 26

Techniques of Dierentiation Theorem (The Product Rule) If f (x) and g(x) are dierentiable at x, then so their product f (x) g(x) and d dx [f (x) g(x)] = d dx [f (x)] g(x) + f (x) d [g(x)] dx or equivalently (fg) = f g + fg In words, the derivative of the product fg is g times the derivative of f plus f times the derivative of g. Theorem (The Quotient Rule) If f (x) and g(x) are dierentiable at x, then so is the quotient Q(x) = f (x) g(x) and d dx [ ] f (x) = g(x) d dx [f (x)] g(x) + f (x) d dx [g(x)] [g(x)] 2 provided g(x) 0 or equivalently ( ) f = f g fg g g 2 13 / 26

Techniques of Dierentiation Denition (The nth derivative) For any positive integer n, the nth derivative of a function is obtained from the function by dierentiating successively n times. If the original function is y = f (x), the nth derivative is denoted by f (n) (x) or d n y dx n 14 / 26

Techniques of Dierentiation Theorem (The Chain Rule) If y = f (u) is dierentiable function of u and u = g(x) is dierentiable function of x, then the composite function y = f (g(x)) is a dierentiable function of x whose derivative is given by the product dy = dy du dx du dx or, equivalently, by dy = dx f (g(x)) g (x) One way to remember the chain rule is to pretend the derivative dy du fractions and to divide out du; that is dy = dy du dx du dx du and dx are 15 / 26

Techniques of Dierentiation Theorem (The General Power Rule) For any real number n and dierentiable function h, d dx [h(x)]n = n [h(x)] n 1 d dx [h(x)] 16 / 26

Marginal Analysis In economics, the use of the derivative to approximate the change in a quantity that results from 1 unit increase in production is called marginal analysis. For instance, suppose C(x) is the total cost of producing x units of particular commodity. If x 0 units are currently being produced, then the derivative C C(x 0 + h) C(x 0) (x 0) = lim h 0 h is called the marginal cost if producing x 0 units. Notice, that if we take h = 1 and x 0 is large in relation to h = 1 we have that C(x 0 + h) C(x 0) 1 = C(x 0 + h) C(x 0) C (x 0) 17 / 26

Marginal Analysis Marginal Revenue and Marginal Prot Suppose R(x) is the revenue generated when x units of a particular commodity are produced, and P(x) is the corresponding prot. When x = x 0 units are being produced, then: The marginal revenue is R (x 0). It approximates R(x 0 + 1) R(x 0), the additional revenue generated by producing one more unit. The marginal prot is P (x 0). It approximates P(x 0 + 1) P(x 0), the additional provit generated by producing one more unit. 18 / 26

Marginal Analysis Example (Using Marginal Analysis to Make a Business Decision) Quentin is the business manager for a company that manufactures digital cameras. He determines that hen x hundred cameras are produced, the total prot will be P(x) thousand dollars where P(x) = 0.0035x 3 + 0.07x 2 + 25x 200 Quentin plans to use marginal prot to make decisions regarding future production. a) What is the marginal prot function? P (x) = 0.0035(3x 2 ) + 0.07(2x) + 25 = 0.0105x 2 + 0.14x + 25 b) The current level of production is x = 10 (1, 000 cameras). Based on the marginal prot at this level of production, should Quentin recommend increasing or decreasing production to increase prot? We have P (10) = 23.35 1-unit increase in production from 10 to 11 hundred cameras increases prot by approximately 23,350 dollars. c) What decision should Quentin make if the current level of production is x = 50 (5, 000 cameras)? What if x = 80 (8, 000 cameras)? 23 / 26

Thank you for your attention! 26 / 26