Chapter 9. Derivatives. Josef Leydold Mathematical Methods WS 2018/19 9 Derivatives 1 / 51. f x. (x 0, f (x 0 ))

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1 Chapter 9 Derivatives Josef Leydold Mathematical Methods WS 208/9 9 Derivatives / 5 Difference Quotient Let f : R R be some function. The the ratio f = f ( 0 + ) f ( 0 ) = f ( 0) 0 is called difference quotient. (, ) secant ( 0, f ( 0 )) f 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 2 / 5 Differential Quotient If the it f ( 0 + ) f ( 0 ) f ( 0 ) = eists, then function f is called differentiable at 0. This it is then called differential quotient or (first) derivative of function f at 0. We write f ( 0 ) or d f d =0 Function f is called differentiable, if it is differentiable at each point of its domain. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 3 / 5

2 Slope of Tangent The differential quotient gives the slope of the tangent to the graph of function at 0. secants f ( 0 ) tangent 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 4 / 5 Marginal Function Instantaneous change of function f. Marginal function (as in marginal utility) 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 5 / 5 Eistence of Differential Quotient Function f is differentiable at all points, where we can draw the tangent (with finite slope) uniquely to the graph. Function f is not differentiable at all points where this is not possible. In particular these are jump discontinuities kinks in the graph of the function vertical tangents Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 6 / 5

3 Computation of the Differential Quotient We can compute differential quotient by determining the it of the difference quotient. Let = 2. The we find for the first derivative f ( 0 + h) ( 0 ) = h 0 h = h + h h 0 h = h h + h 2 h = 2 0 = h 0 (2 0 + h) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 7 / 5 Derivative of a Function Function f : D R, f () = d f d is called the first derivative of Function f. Its domain D is the set of all points where the differential quotient (i.e., the it of the difference quotient) eists. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 8 / 5 Derivatives of Elementary Functions f () c 0 α e ln() sin() cos() α α e cos() sin() Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 9 / 5

4 Computation Rules for Derivatives (c ) = c f () ( + g()) = f () + g () ( g()) = f () g() + g () ( f (g())) = f (g()) g () ( ) = f () g() g () g() (g()) 2 Summation rule Product rule Chain rule Quotient rule Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 0 / 5 Eample Computation Rules for Derivatives ( ) = = ( e 2) = (e ) 2 + e ( 2) = e 2 + e 2 ( (3 2 + ) 2) = 2 (3 2 + ) 6 ( ) = ( 2 ) = 2 2 = 2 (a ) = ( e ln(a) ) = e ln(a) ln(a) = a ln(a) ( ) + 2 = 2 ( 3 ) ( + 2 ) ( 3 ) 2 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives / 5 Higher Order Derivatives We can compute derivatives of the derivative of a function. Thus we obtain the second derivative f () of function f, third derivative f (), etc., n-th derivative f (n) (). Other notations: d 2 f d 2, d 3 f d 3,..., d n f d n Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 2 / 5

5 Eample Higher Order Derivatives The first five derivatives of 5 function = are f () = ( ) = f () = ( ) = f () = ( ) = 24 f ıv () = (24) = 24 f v () = 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 3 / 5 Marginal Change We can estimate the derivative f ( 0 ) approimately by means of the difference quotient with small change : f f ( 0 + ) f ( 0 ) ( 0 ) = f 0 Vice verse we can estimate the change f of f for small changes approimately by the first derivative of f : Beware: f = f ( 0 + ) f ( 0 ) f ( 0 ) f ( 0 ) is a linear Function in. It is the best possible approimation of f by a linear function around 0. This approimation is useful only for small values of. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 4 / 5 Differential The approimation f = f ( 0 + ) f ( 0 ) f ( 0 ) becomes eact if (and thus f ) becomes infinitesimally small. We then write d and d f instead of and f, resp. d f = f ( 0 ) d Symbols d f and d are called the differentials of function f and the independent variable, resp. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 5 / 5

6 Differential Differential d f can be seen as a linear function in d. We can use it to compute f approimately around 0. f ( 0 + d) f ( 0 ) + d f Let = e. Differential of f at point: d f = f () d = e d Approimation of f (.) by means of this differential: = ( 0 + d) 0 =. = 0. f (.) f () + d f = e + e Eact value: f (.) = Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 6 / 5 Elasticity The first derivative of a function gives absolute rate of change of f at 0. Hence it depends on the scales used for argument and function values. However, often relative rates of change are more appropriate. We obtain scale invariance and relative rate of changes by and thus change of function value relative to value of function change of argument relative to value of argument 0 f (+ ) f ( + ) = 0 = f () Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 7 / 5 Elasticity The epression ε = f () is called the elasticity of f at point. Let = 3 e 2. Then Let = β α. Then ε = f () = 6 e2 3 e 2 = 2 ε = f () = β α α β α = α Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 8 / 5

7 Elasticity II The relative rate of change of f can be epressed as ln( ) = f () What happens if we compute the derivative of ln( ) w.r.t. ln()? Let v = ln() = e v Derivation by means of the chain rule yields: d(ln( )) d(ln()) = d(ln( f (ev ))) dv = f (e v ) f (e v ) ev = f () = ε ε = d(ln( )) d(ln()) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 9 / 5 Elasticity II We can use the chain rule formally in the following way: Let u = ln(y), y =, = e v v = ln() Then we find d(ln f ) d(ln ) = du dv = du dy dy d d dv = y f () e v = f () Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 20 / 5 Elastic Functions A Function f is called elastic in, if ε > -elastic in, if ε = inelastic in, if ε < For elastic functions we then have: The value of the function changes relatively faster than the value of the argument. Function = 3 e 2 is [ ε = 2 ] -elastic, for = 2 and = 2 ; inelastic, for 2 < < 2 ; elastic, for < 2 or > 2. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 2 / 5

8 Source of Errors Beware! Function f is elastic if the absolute value of the elasticity is greater than. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 22 / 5 Elastic Demand Let q(p) be an elastic demand function, where p is the price. We have: p > 0, q > 0, and q < 0 (q is decreasing). Hence ε q (p) = p q (p) q(p) < What happens to the turnover (= price selling)? u (p) = (p q(p)) = q(p) + p q (p) = q(p) ( + p q (p) ) q(p) }{{} =ε q < < 0 In other words, the turnover decreases if we raise prices. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 23 / 5 Partial Derivative We investigate the rate of change of function f (,..., n ), when variable i changes and the other variables remain fied. f f (..., = i + i,...) f (..., i,...) i i 0 i is called the (first) partial derivative of f w.r.t. i. There eist a other symbols that denote the partial derivative f i : f i () (derivative w.r.t. variable i ) f i () f i () (derivative w.r.t. the i-th variable) (i-th component of the gradient) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 24 / 5

9 Computation of Partial Derivatives We obtain partial derivatives f when we use the rules for univariate i functions for variable i while we treat all other variables as constants. First partial derivatives of f (, 2 ) = sin(2 ) cos( 2 ) f = 2 cos(2 ) cos( 2 ) }{{} treated as constant f 2 = sin(2 ) }{{} treated as constant ( sin( 2 )) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 25 / 5 Higher Order Partial Derivatives We can compute partial derivatives of partial derivatives analogously to their univariate counterparts and obtain higher order partial derivatives: 2 f k i () and 2 f () 2 i There eist a other symbols that denote the partial derivatives f i k () (derivative w.r.t. variables i and k ) f ik () f ik () (derivative w.r.t. the i-th and k-th variable) (component of the Hessian matri with inde ik) 2 f k i (): Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 26 / 5 Higher Order Partial Derivatives If all second order partial derivatives eists and are continuous, then the order of differentiation does not matter (Schwarz s theorem): 2 f k i () = 2 f i k () Remark: Practically all differentiable functions in economic models have this property. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 27 / 5

10 Eample Higher Order Partial Derivatives Compute the first and second order partial derivatives of f (, y) = y First order partial derivatives: f = y f y = Second order partial derivatives: f = 2 f y = 3 f y = 3 f yy = 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 28 / 5 Gradient We collect all first order partial derivatives into a (row) vector which is called the gradient at point. = ( f (),..., f n ()) read: gradient of f or nabla f. Other notation: f () The gradient also can be a column vector. The gradient is the analog of the first derivative of univariate functions. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 29 / 5 Properties of the Gradient The gradient always of f always points to the direction of steepest ascent. Its length is equal to the slope at this point. The gradient is normal (i.e. in right angle) to the corresponding contour line (level set). f Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 30 / 5

11 Eample Gradient Compute the gradient of f (, y) = y at point = (3, 2). f = y f y = = (2 + 3y, 3) f (3, 2) = (2, 9) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 3 / 5 Directional Derivative We obtain partial derivative f by differentiating the univariaten i function g(t) = f (,..., i + t,..., n ) = f ( + t h) with h = e i at point t = 0: f () = dg i dt f 2 t=0 = d dt f ( + t h) t=0 f Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 32 / 5 Directional Derivative Generalization: We obtain the directional derivative f h along h with length by differentiating the univariaten function g(t) = f ( + t h) at point t = 0: f dg () = h dt = d t=0 dt t=0 f ( + t h) f h h The directional derivative describes the change of f, if we move in direction h. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 33 / 5

12 Directional Derivative For have (for h = ): f h () = f () h + + f n () h n = h If h does not have norm, we first have to normalize first: f h () = h h Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 34 / 5 Eample Directional Derivative Compute the directional derivative of f (, 2 ) = ( ) ( ) 3 along h = at =. 2 2 Norm of h: h = h t h = 2 + ( 2) 2 = 5 Directional derivative: ( ) f h () = h h = (2, 9) 5 2 = 6 5 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 35 / 5 Total Differential We want to approimate a function f by some linear function so approimieren such that the approimation error is as small as possible: f ( + h) f () h f n () h n = h The approimation becomes eact if h (and thus f ) becomes infinitesimally small. The linear function d f = f () d f n () d n = is called the total Differential of f at. n f i d i = d i= Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 36 / 5

13 Eample Total Differential Compute the total differential of at = (3, 2). f (, 2 ) = d f = f (3, 2) d + f 2 (3, 2) d 2 = 2 d + 9 d 2 Approimation of f (3.,.8) by means of the total differential: f (3.,.8) f (3; 2) + d f = ( 0.2) = Eact value: f (3.,.8) = ( ) ( ) ( ) h = ( + h) = = Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 37 / 5 Hessian Matri Let = f (,..., n ) be two times differentiable. Then matri f () f 2 ()... f n () f H = 2 () f 2 2 ()... f 2 n () f n () f n 2 ()... f n n () is called the Hessian matri of f at. The Hessian matri is symmetric, i.e., f i k () = f k i (). Other notation: f () The Hessian matri is the analog of the second derivative of univariate functions. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 38 / 5 Eample Hessian Matri Compute the Hessian matri of at point = (, 2). Second order partial derivatives: f (, y) = y f = 2 f y = 3 f y = 3 f yy = 0 Hessian matri: ( ) ( ) f (, y) f y (, y) 2 3 H f (, y) = = = H f (, 2) f y (, y) f yy (, y) 3 0 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 39 / 5

14 Differentiability Theorem: A function f : R R is differentiable at 0 if and only if there eists a linear map l which approimates f in 0 in an optimal way: Obviously l(h) = f ( 0 ) h. ( f ( 0 + h) f ( 0 )) l(h) = 0 h 0 h Definition: A function f : R n R m is differentiable at 0 if there eists a linear map l which approimates f in 0 in an optimal way: (f( 0 + h) f( 0 )) l(h) = 0 h 0 h Function l(h) = J h is called the total derivative of f. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 40 / 5 Jacobian Matri f (,..., n ) Let f : R n R m, y = f() =. f m (,..., n ) The m n matri f Df( 0 ) = f... ( 0 ) =..... f m... is called the Jacobian matri of f at point 0. f n f m n It is the generalization of derivatives (and gradients) for vector-valued functions. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 4 / 5 Jacobian Matri For f : R n R the Jacobian matri is the gradient of f : D f ( 0 ) = f ( 0 ) For vector-valued functions the Jacobian matri can be written as f ( 0 ) Df( 0 ) =. f m ( 0 ) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 42 / 5

15 Eample Jacobian Matri = f (, 2 ) = ep( ( ) ) D = f, f 2 = = ( 2 ep( ), 2 2 ep( )) ( ) ( f (, 2 ) 2 f() = f(, 2 ) = = + ) 2 2 f 2 (, 2 ) ( f f ) ( ) Df() = f 2 = f 2 2 ( ) ( ) s (t) cos(t) s(t) = = s 2 (t) sin(t) ) ( ) sin(t) Ds(t) = = cos(t) ( ds dt ds 2 dt Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 43 / 5 Chain Rule Let f : R n R m and g : R m R k. Then (g f) () = g (f()) f () ( ) ( ) e 2 + y 2 f(, y) = e y g(, y) = 2 y 2 ( ) ( ) f e 0 (, y) = 0 e y g 2 2 y (, y) = 2 2 y ( ) ( ) (g f) () = g (f()) f 2 e 2 e y e 0 () = 2 e 2 e y 0 e ( ) y 2 e 2 2 e 2y = 2 e 2 2 e 2y Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 44 / 5 Eample Directional Derivative We can derive the formula for the directional derivative of f : R n R along h (with h = ) at 0 by means of the chain rule: Let s(t) be a path in along h, i.e., Then and hence s : R R n, t 0 + th. f (s(0)) = f ( 0 ) = f ( 0 ) s (0) = h f h = ( f s) (0) = f (s(0)) s (0) = f ( 0 ) h. Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 45 / 5

16 Eample Indirect Dependency Let f (, 2, t) where (t) and 2 (t) also depend on t. What is the total derivative of f w.r.t. t? Chain rule: (t) Let : R R 3, t 2 (t) t d f dt = ( f ) (t) = f ((t)) (t) (t) = f ((t)) 2 (t) (t) = ( f ((t)), f 2 ((t)), f t ((t)) 2 (t) = f ((t)) (t) + f 2 ((t)) 2 (t) + f t((t)) = f (, 2, t) (t) + f 2 (, 2, t) 2 (t) + f t(, 2, t) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 46 / 5 L Hôpital s Rule Suppose we want to compute 0 g() and find = g() = (or = ± ) However, epressions like 0 0 or are not defined. (You must not reduce the fraction by 0 or!) Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 47 / 5 L Hôpital s Rule If 0 = 0 g() = 0 (or = or = ), then 0 g() = 0 f () g () Assumption: f and g are differentiable in 0. This formula is called l Hôpital s rule (also spelled as l Hospital s rule). Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 48 / 5

17 Eample L Hôpital s Rule = = 5 3 ln 2 = 2 = 2 2 = 0 ln( + ) ( + ) 0 2 = 0 2 = 0 ( + ) 2 2 = 2 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 49 / 5 Eample L Hôpital s Rule L Hôpital s rule can be applied iteratively: e e 0 2 = e = = 2 Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 50 / 5 Summary Difference quotient and differential quotient Differential quotient and derivative Derivatives of elementary functions Differentiation rules Higher order derivatives Total differential Elasticity Partial derivatives Gradient and Hessian matri Jacobian matri and chain rule L Hôpital s rule Josef Leydold Mathematical Methods WS 208/9 9 Derivatives 5 / 5