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Lecture 1 Differentiation Agnieszka Kulacka Birkbeck College, University of London

Differential and Integral Calculus Lecture 1: Differentiation Lecture 2: Unconstrained optimisation Lecture 3: Constrained optimisation Lecture 4: Integration Lecture 5: Difference and differential equations

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Motivation Differential calculus is concerned with the way in which a value of the function changes when the independent variable changes. In economics, we study, for example, how a change in a firm’s output level affects its costs and how a change in a country’s money supply affects the rate of inflation.

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The ratio of ∆y/∆x as ∆x → 0 measures the instantaneous rate of change of y with respect to x, and it’s called the derivative. dy ∆y = lim ∆x→0 ∆x dx The derivative of the function y = f (x) at the point P = (x0, y0) is the slope of the tangent line at that point. f (x0 + ∆x) − f (x0) ∆x→0 ∆x

f ′(x0) = lim

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Example 1. f (x) = x2

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Differential If f ′(x0) is the derivative of the function y = f (x) at the point P (x0, f (x0)), then the total differential at the point P is dy = df (x0, dx) = f ′(x0)dx Thus the differential is a function of both x and dx. The differential provides us with a method of estimating the effect of a change in x of the amount dx = ∆x on y, where ∆y is the exact change while dy is the approximate change in y . This can be expressed as ∆y = dy + ǫ 5

Hoy et al. Mathematics for Economics p. 136 6

Hoy et al. Mathematics for Economics p. 137 7

Hoy et al. Mathematics for Economics p. 137 8

Example 2. Let f (x) = −x2. Use the differential to estimate the changes in y between P = (10, −100) and each of the 5 points Qn, n = 1, 2, 3, 4, 5. Qi

(15,-225)

(14,-196)

(13,-169)

(12,-144)

(11,-121)

∆x = dx ∆y dy = f ′ (x)dx ǫ

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A function f (x), which is defined on an open interval including the point x = a, is differentiable at that point if

f (a + ∆x) − f (a) ∆x→0 ∆x lim

exists and is finite. That is,

f (a + ∆x) − f (a) f (a + ∆x) − f (a) = lim ∆x ∆x ∆x→0− ∆x→0+ lim

The value of this expression is the value of the derivative function f ′(x) at the point x = a. 10

Example 3. Find the left- and right-hand derivatives at the point of nondifferentiability. f (x) =

 x,

2 − x,

x f (x2) whenever x1 < x2. In either case f is said to be a monotonic function. Test for monotonicity If f ′(x) ≥ 0 for all x, and f ′(x) = 0 for only a finite number of values of x, then f is strictly increasing. Similarly, if f ′(x) ≤ 0 for all x, and f ′(x) = 0 for only a finite number of values of x, then f is strictly decreasing. 18

Example 7. Find the interval where the following functions are increasing: (a)

y = (ln x)2 − 4

(c)

3 ln(x2 + 2) y =x−2

(b)

y = ln(ex + e−x)

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Partial Derivatives We will extend the idea of the derivative for functions of one variable to functions defined on Rn. Let y = f (x1, x2, ...xn). We can define the rate at which y changes with respect to changes in each of the variables x1, x2, ...xn, taken separately in the same way as for one independent variable. The partial derivative fi(x1, ..., xi, ..., xn) of a function y = f (x1, x2, ...xn) with respect to variable xi is

∂f f (x1, ..., xi + ∆xi, ..., xn) − f (x1, ..., xi, ..., xn) = lim ∆xi →0 ∆xi ∂xi 20

Hoy et al. Mathematics for Economics p. 395 21

Example 8. ∂z Find ∂z ∂x and ∂y for the following functions

(a)

z = x2 + 3y 2

(b)

z = xy

(c)

z = 5x4y 2 − 2xy 5

(d)

z = ex+y

(e)

z = exy

(f)

z = ex/y

(g)

z = ln(x + y)

(h)

z = ln(xy )

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Gradient Vector Assume that f (x1, ..., xn) is continuous and its partial derivatives are defined for all (x1, ...xn). Then we may define the gradient vector   f1    f2  ∇f =    ...  fn ∂f . where fi = ∂x i

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Example 9. Find the gradient vector for the following functions (a)

f (x, y) = x7 − y 7

(b)

f (x, y) = x5 ln y

(c)

f (x, y) = (x2 − 2y 2)5

(d)

f (x, y, z) = x2 + y 3 + z 4

(e)

f (x, y, z) = xyz

(f)

f (x, y, z) = x4/yz

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Total differential 



dx1   Let dx =  ... . dxn The first-order total differential for the function y = f (x1, ..., xn) is dy = f1dx1 + f2dx2 + ... + fndxn = (∇f )T dx ∂f where fi = ∂x . i

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Example 10. Calculate the differentials of the following functions (a)

z = x3 + y 3

(b)

z = xey

2

(c)

z = ln(x2 − y 2)

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Using the total differential to approximate the actual change ∆y in the function value for given changes in x1, ..., xn corresponds geometrically to using the tangent plane as an approximation of the function in the same way as we used the tangent line for the functions of one variable.

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Hoy et al. Mathematics for Economics p. 417 28

Example 11. Let T (x, y, z) = (x2 + y 2 + z 2)1/2. Find the total differential and use it to estimate the changes in T if x changes from 2 to 2.01 and y changes from 3 to 2.99 and z changes from 6 to 6.02.

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Total derivative Using a chain rule, we can define a total derivative

d dx f (x1, x2, ..., xn) = (∇f )T dt dt

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Example 12. Find dz dt when (a) z = x2 + y 3 with x = t2 and y = 2t (b) z = xe2y with x =

√ t and y = ln t

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