RS - Ch 7 - Rules of Differentiation Chapter 7 Rules of Differentiation & Taylor Series 1 Isaac Newton and Gottfried Leibniz 7.1 Review: Derivative and Derivative Rules • Review: Definition of derivative. y f x x f x dy f '(x) lim lim 0 0 x0 x x0 x dx • Applying this definition, we review the 9 rules of differentiation: 1) Constant: 0 2) Power: 3) Sum/Difference ∗ 4) Product 2 1 RS - Ch 7 - Rules of Differentiation 7.1 Review: Derivative and Derivative Rules • (continuation) 9 rules of differentiation: / 5) Quotient (from 4) / 6) Exponential 7) Chain Rule ∗ (with 8) Inverse function. Let be a strictly monotonic function. . 9) Constant, Product and Power (from 1, 2 & 4) 3 7.1.1 Constant Rule • Recall the definition of derivative. y f x x f x dy f '(x) lim lim 0 0 x0 x x0 x dx • Applying this definition, we derive the constant rule: The derivative of a constant function is zero for all values of x. dy d y f x k k 0 dx dx dy f ( x x) f ( x) f '(x) lim dx x 0 x If f(x) k then f(x x) k k k lim lim 0 0 x 0 x x 0 4 2 RS - Ch 7 - Rules of Differentiation 7.1.2 Power-Function Rule f x Δx f x (x Δx)n x n f '(x) lim lim x0 Δx Δx0 Δx (x n nx n1Δx (n 1)x n2Δx 2 ... nxΔx n1 Δx n ) x n lim x0 Δx lim x n / Δx nx n1 (n 1)x n2Δx ... nxΔx n2 Δx n1 x n / Δx x0 nx n1 (x Δx)n x n lim nx n1 Δx0 Δx Example: Let Total Revenue (R) be: R = 15 Q – Q2 15 2. As Q increases R increases (as long as Q > 7.5). 5 7.1.3 Sum or Difference Rule d 3) f x g x f x g x dx • The derivative of a sum (or difference) of two functions is the sum (or difference) of the derivatives of the two functions Example : C Q 3 4Q 2 10Q 75 dC d d d d Q3 4Q 2 10Q 75 dQ dQ dQ dQ dQ dC 3Q 2 8Q 10 0 dQ 6 3 RS - Ch 7 - Rules of Differentiation 7.1.4 Product Rule ∗ 4 The derivative of the product of two functions is equal to the second function times the derivative of the first plus the first function times the derivative of the second. Example: Marginal Revenue (MR) Total Revenue: R = P Q Given 15 R 15 1∗ 15 15 2 Same as in previous example. 7 7.1.5 Quotient Rule / 5 / Example : TC C(Q) Total cost AC C(Q)/Q Average cost d CQ Q CQ C Q 1 1 CQ 1 2 C Q MC AC dQ Q Q Q Q Q d CQ if 0, then AC MC (Average Cost Marginal Cost) dQ Q 8 4 RS - Ch 7 - Rules of Differentiation 7.1.6 Exponential-Function Rule f x Δx f x e (xΔx) ex (eΔx 1) f '(x) lim lim ex lim x0 Δx Δx0 Δx x0 Δx (eh 1) Definition of e : e unique positive number for which lim 1 h0 h (eh 1) Let h κΔx. Then, lim κ h0 h e (xΔx) ex Thus, lim ex Δx0 Δx • Example : Exponential Growth d y f t e0.5t e0.5t 0.5e0.5t dt 9 7.1.6 Exponential-Function Rule: Joke A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!" One day he met a new patient; and true to form he stared at him and said "I differentiate you!", but for once, his victim's expression didn't change. Surprised, the mathematician collected all his energy, stared fiercely at the new patient and said loudly "I differentiate you!", but still the other man had no reaction. Finally, in frustration, the mathematician screamed out "I DIFFERENTIATE YOU!" The new patient calmly looked up and said, "You can differentiate me all you like: I'm ex." 5 RS - Ch 7 - Rules of Differentiation 7.1.7 Chain Rule This is a case of two or more differentiable functions, in which each has a distinct independent variable, where z f(g(x)). That is, z f(y), i.e., z is a function of variable y and y g(x), i.e., y is a function of variable x dz dz dy 7) dx dy dx df y df y dgx f y g x dx dy dx Example: R f(Q) (revenue) & Q g(L) (output) dR dR dQ dL dQ dL f Q g L 11 MR MPPL MRPL 7.1.7 Chain rule: Application – Log rule dz dz dy • Chain Rule : dx dy dx df y df y dg x f y g x dx dy dx • Consider h(x) = eln(x) = x. h’(x) = 1. Now, apply Chain rule to h(x): d ln( x) d ln( x) h'(x) e ln( x ) x dx dx d ln( x) d ln( x) 1 1 x dx dx x 12 6 RS - Ch 7 - Rules of Differentiation 7.1.8 Inverse-function Rule • Let y = f(x) be a differentiable strictly monotonic function: . Note: A monotonic function is one in which a given value of x yields a unique value of y, and given a value of y will yield a unique value of x (a one-to-one mapping). These types of functions have a defined inverse. Example: Inverse supply function P (where > 0) 13 7.1.8 Inverse-function Rule This property of one-to-one mapping is unique to the class of functions known as monotonic functions: Recall the definition of a function: function: one y for each x monotonic function: one x for each y (inverse function) if x1 > x2 f(x1) > f(x2) monotonically increasing Qs = b0 + b1P supply function (where b1 > 0) P = – b0/b1 + (1/b1)Qs inverse supply function if x1 > x2 f(x1) < f(x2) monotonically decreasing Qd = a0 –a1P demand function (where a1 > 0) P = a0/a1 –(1/a1)Qd inverse demand function 14 7 RS - Ch 7 - Rules of Differentiation 7.2 Extension: Partial Differentiation In multivariate calculus, y depends on several variables: y = f(x1, x2, …, xn.) The derivative of y with respect to one of the variables –while the other variables are held constant– is called a partial derivative. y f (x1, x2 ,⋯ , xn ) y f (x x , x ,⋯ , x ) f (x , x ,⋯ , x ) lim lim 1 1 2 n 1 2 n x1 0 x1 x1 0 x1 y f1 (partial derivative w.r.t. x1) x1 y y In general, lim fi , i 1...n xi 0 xi xi 15 7.2 Application: Black-Scholes – Greeks The Black-Scholes (BS) formula price an European call option on a non-dividend paying stock, as a function of the stock price (St), time to maturity (T-t), strike price (K), interest rates (i) and the stock price volatility (): 1 2 where (standard normal distribution function) 2 d1 = [ln(St/K) + (i + /2) (T – t)]/( ), 2 d2 = [ln(St/K)+(i– /2) (T – t)]/( )) = d1 – • The Greeks represent the first derivatives of the BS pricing formulas (ceteris paribus) with respect to the driver variables: St, (T-t), i, .For 16 example, the first derivative with respect to St is called Δ (or BS Delta). 8 RS - Ch 7 - Rules of Differentiation 7.2 Application: Black-Scholes – Greeks (Delta) • Δ = BS Delta Δ = 1 ∗ ∗ • Taking derivatives and using the FTC to get N'(d): 1 2 Then, 1 1 1 1 Δ 1 ∗ ∗ 2 2 17 7.2 Application: Black-Scholes – Greeks (Delta) Δ 1 ′ 1 ∗ ∗ ] 1 (from definition of d1) since • We can use Δ to establish a portfolio that is not sensitive to changes in St: A long position in one call and a short position Δ stocks. The profits from this portfolio are: Π = Ct – Δ St Then, Δ= 0. Note: A position with a delta of zero is referred to as being delta-neutral. 18 9 RS - Ch 7 - Rules of Differentiation 7.2 Application: Black-Scholes – Greeks (Vega) • V = BS Vega: It measures the sensitivity of option prices to changes in volatility. Recall BS formula: 1 2 Then, V= ′1 ∗ ′2 ∗ Using the result (check it): = + and after some algebra, similar to what we did above with the result: ′ 1 ′ 2 , we get to: V 1 0 19 7.2 Application: Black-Scholes – Greeks (Rho) • P = BS Rho: It measures the sensitivity of option prices to changes in interest rates. For the call option we get: 1 2 P 1 ∗ 2 ∗ 2 Using the result (check it): = A bit of algebra, again, mainly using: ′ 1 ′ 2 delivers P ∗2 20 10 RS - Ch 7 - Rules of Differentiation 7.3 Second & Higher Derivatives Derivative of a derivative Given y = f(x) The first derivative f '(x) or dy/dx is itself a function of x, it should be differentiable with respect to x, provided that it is continuous and smooth. The result of this differentiation is known as the second derivative of the function f and is denoted as f ''(x) or d2y/dx2. The second derivative can be differentiated with respect to x again to produce a third derivative: f '''(x) and so on to f(n)(x) or dny/dxn This process can be continued to produce an n-th derivative.