Mathematics in Economics
Lecture 2
Mgr. Jiří Mazurek, Ph.D.
Mathematics in Economics/PMAT

Lecture 2
Introduction to differential calculus of one real variable
The derivative of a function

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Let y = f (x) be a function of one real variable. Then the
 derivative of the function f is defined as follows:
The derivative is usually denoted f´(x) or y´. The process of finding a derivative is called
differentiation.
Geometric interpretation: the derivative of a function f at a point x is equal to the slope of a
tangent line to the curve at the point x.

Lecture 2
Geometric interpretation of a derivative

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Lecture 2
The rules of differentiation

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Let f(x) and g(x) be functions with the derivative
in the interval               Then:

Lecture 2
Derivatives of elementary functions

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Lecture 2
Examples

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Lecture 2
Derivatives of higher orders

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If a function f´(x) can be differentiated, we obtain
the second derivative of f(x), denoted as f´´(x), and so on.
First derivatives are used to find monotonicity and extremas of functions. The second derivative is
useful in finding concavity and concavity or inflexion points. The use of derivatives of the order
3 and higher are rather rare.
Example:

Lecture 2
Differential of a function
 The differential of a function y = f(x) is denoted as dy, and is defined as
follows:                    .
The differential expresses an increment of the dependent variable dy in respect to the increment of
independent variable dx. Also, the differential isused to linearization of more complex functions.
Example: Find the the differential of the function             at a point x = 4.
Solution: dy=2xdx, and substituting x = 4 we obtain: dy = 8dx.

Lecture 2
The logarithmic differentiation
   For functions of the type                        we use the so called logarithmic
differentiation:

Lecture 2
Taylor and Maclaurin series
Let a function y = f(x) be differentiable of the order n at a point a, then it can be approximated
by the Taylor series of the form:
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•If a = 0, we obtain a special case of the Taylor series, called Maclaurin series:
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Lecture 2
Maclaurin series of selected functions

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Function
Maclaurin series
Range of convergence
sinx
cosx
exp(x)
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Lecture 2
Taylor and Maclaurin series
 Example: Find the Maclaurin series of the function           .
Solution:
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Therefore, we obtain:
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Lecture 2
Economic applications of derivatives
 The elasticity of a function y = f(x):
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 The price elasticity of demand:
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 The price elasticity of supply:
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Lecture 2
Economic applications of derivatives – cont.
 Marginal product of labour:
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 Marginal revenue:
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  Marginal cost:
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Lecture 2
Solved problems 1

   Find the derivative of the function
  Solution: The given function is a product of two elementary functions, so we must use the product
rule.
  We obtain:

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Lecture 2
Solved problems 2

Find the differential of the function                   . Also, find the increment dy for x = 2 and
dx = 0.1
Solution:             and the differential is:
The increment dy:

Lecture 2
Solved problems 3
 Find the Taylor series of the function            at the point a = 1.
Solution:

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Therefore, the Taylor series is given as follows:
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Lecture 2
Solved problems 4
 Find the Maclaurin series of the function                 .
Solution:

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Therefore, the Taylor series is given as follows:
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Lecture 2
Solved problems 5
 Find the derivative of the function           .
Solution:

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Lecture 2
Solved problems 6
 Find the derivative of the function             .
Solution:

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Lecture 2
Solved problems 7
 Find marginal revenue MR (x) of the total revenue
                                     and marginal costs of the total costs
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Solution:
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Lecture 2
Problems to solve 1
 Differentiate:
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Lecture 2
Problems to solve 2
 Differentiate:
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Lecture 2
Problems to solve 3
 Find the Maclaurin series of the following functions:
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Lecture 2
Thank you for your attention!

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