Mathematics 2 for Business Schools Section 1: Fundamentals ...

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Mathematics 2 for Business Schools

Section 1: Fundamentals of Differential Calculus

Spring Semester 2017

After finishing this section you should be able to …

• derive the difference quotient and the differential quotient of a function (repetition).

• explain the concept of the derivative of functions (repetition).

• derive and correctly apply the rule for the derivative of constant functions (repetition).

• derive and correctly apply the rule for the derivative of power functions (repetition).

• correctly apply the constant factor rule and the sum rule (repetition).

• correctly apply the product rule, the quotient rule, and the chain rule (new).

• find the derivative of exponential functions and logarithmic functions (new).

Learning objectives

2Spring semester 2017 Section 1: Fundamentals of differential calculus

Difference quotient – Definition

The slope 𝑚𝑠 of the secant through the

points

𝑃 𝑥0, 𝑓(𝑥0) and

𝑄 𝑥0 + Δ𝑥, 𝑓(𝑥0 + Δ𝑥)

of 𝑓, i.e., the average rate of change

of 𝑓 on the interval 𝑥0; 𝑥0 + Δ𝑥 is

called the difference quotient

𝑚𝑠 =Δ𝑓

Δ𝑥=𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0

Δ𝑥

at 𝑥0 (between 𝑃 and 𝑄).

Spring semester 2017 Section 1: Fundamentals of differential calculus

𝑥0 𝑥0 + ∆𝑥

𝑓(𝑥0)

𝑓(𝑥0 + ∆𝑥)

Δ𝑓

Δ𝑥

Local rate of change of a function

𝑚𝑡 = limΔ𝑥⟶0

𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0Δ𝑥

Finally, the secant becomes the tangent and the

slope of the secant becomes the slope of the

tangent.

The average rate of change of a function between 𝑃 and 𝑄 becomes the local rate of

change in 𝑃 if 𝑄 is moved towards 𝑃, i.e. if ∆𝑥 tends to 0.

Differential quotient – Definition

The function 𝑓 is called differentiable in 𝑥0 if the

limit of the difference quotient

𝑚𝑡 = 𝑓′ 𝑥0 = limΔ𝑥⟶0

𝑓 𝑥0+Δ𝑥 −𝑓 𝑥0

Δ𝑥

exists in 𝑥0.

Our notation for this limit is 𝑓′ 𝑥0 and we call it

differential quotient,

derivative,

slope of the tangent or

local rate of change of 𝑓

in 𝑥0.

𝑓(𝑥0)

There are several notations used for the derivative.

The most widely used notation for the derivative of 𝑓 is 𝑓′. This notation was introduced

by Newton.

Since the derivative is the same as the differential quotient, the Leibniz notation is also

used quite often. Here, the derivative of 𝑓 is written as 𝑑𝑓

𝑑𝑥.

Both notations mean the same, namely the derivative of 𝑓. Therefore 𝑓′ =𝑑𝑓

𝑑𝑥.

The calculator can numerically find the derivative at a specific point 𝑥0. This is done by

using the following keys:

SHIFT 𝑑/𝑑𝑥,

enter the function,

enter the 𝑥-value where to calculate 𝑓′.

d-notation and calculator

Differentiation rules – Review

𝑓(𝑥) 𝑓′(𝑥) Remarks

1. 𝑐 0 𝑐 ∈ ℝ, any real constant

2. 𝑥 1

3. 𝑥𝑟 𝑟 ∙ 𝑥𝑟−1

4. 𝑐 ∙ 𝑔(𝑥) 𝑐 ∙ 𝑔′(𝑥) 𝑐 ∈ ℝ, factor rule

5. 𝑢 𝑥 + 𝑣(𝑥) 𝑢′ 𝑥 + 𝑣′(𝑥) sum rule

These rules can be expressed in a shorter way by omitting the argument x, e.g. the sum rule can be written as

𝑢 + 𝑣 ′ = 𝑢′ + 𝑣′ instead of 𝑢 𝑥 + 𝑣 𝑥 ′ = 𝑢′ 𝑥 + 𝑣′(𝑥)

Exercise – Repetition

Find the first and second derivatives of the following functions:

a) 𝑔 𝑥 = 5

b) 𝑝 𝑥 = 𝑥2

c) 𝑓 𝑡 = 3𝑥4

d) 𝑓 𝑡 = 5𝑡3

e) ℎ 𝑢 = 3𝑢4 −1

2𝑢2 + 3

f) 𝑓 𝑥 =3𝑥2 +

𝑥2−

Product rule

If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then 𝑓 𝑥 = 𝑢(𝑥) ∙ 𝑣(𝑥) is also

differentiable and its derivative is given by:

𝑓(𝑥)′ = 𝑢′ 𝑥 ⋅ 𝑣(𝑥) + 𝑢(𝑥) ⋅ 𝑣′(𝑥)

Short notation: 𝑢 ∙ 𝑣 ′ = 𝑢′𝑣 + 𝑢𝑣′

Example: 𝑓 𝑥 = 𝑥 + 1 𝑥 − 1

𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1

𝑓′ 𝑥 = 1 ∙ 𝑥 − 1 + 𝑥 + 1 ∙ 1= 𝑥 − 1 + 𝑥 + 1 = 2𝑥

(One can reach the same result by first expanding the functional term and then differentiating the expression in its expanded form.)

Quotient rule

If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then the function 𝑓 𝑥 =𝑢(𝑥)

𝑣(𝑥)is

also differentiable and its derivative is given by:

𝑓(𝑥)′ =𝑢′(𝑥) ⋅ 𝑣 𝑥 − 𝑢 𝑥 ⋅ 𝑣′(𝑥)

𝑣(𝑥)2

Short notation :𝑢

𝑢′𝑣−𝑢𝑣′

Example: 𝑓 𝑥 =𝑥+1

𝑥−1

𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1

𝑓′ 𝑥 =1∙ 𝑥−1 − 𝑥+1 ∙1

𝑥−1 2 =𝑥−1−𝑥−1

𝑥−1 2 =−2

𝑥−1 2

Note theminus sign!

Chain rule

Example:

𝑓 𝑥 = 3𝑥2 + 6𝑥 79

Outer function: 𝑢 𝑣

Inner function: 𝑣(𝑥)

Is the composition of the two functions 𝑢 ∘ 𝑣 exists and if 𝑢 is

differentiable at 𝑣(𝑥) then derivative of the composition is given by:

𝑢 ∘ 𝑣 ′ 𝑥 = 𝑢 𝑣 𝑥 ′ = 𝑢′ 𝑣 𝑥 ⋅ 𝑣′(𝑥)

𝑓′ 𝑥 = 79 ⋅ 3𝑥2 + 6𝑥 78 ⋅ 6𝑥 + 6 = ⋯

Outer derivative : 𝑢′ 𝑣 Inner derivative: 𝑣′(𝑥)

Short notation : 𝑢 ∘ 𝑢 ′ = 𝑢′(𝑣) ∙ 𝑣′ In other words: outer derivative multiplied by inner derivative

Examples – Product rule, quotient rule, and chain rule

Find the derivatives of

a) 𝑓 𝑥 = 2𝑥2 − 1 3𝑥 + 1

b) 𝑔 𝑥 =2𝑥2−1

3𝑥+1

c) ℎ 𝑥 =32𝑥2 − 1

Derivatives of exponential and logarithmic functions

For 𝑎 ∈ ℝ+\ 1 the functions 𝑎𝑥 and log𝑎𝑥 are differentiable and

their derivatives are:

a) 𝑒𝑥 ′ = 𝑒𝑥

b) 𝑎𝑥 ′ = 𝑎𝑥 ⋅ ln 𝑎

c) ln 𝑥 ′ =1

d) log𝑎 𝑥′ =

𝑥⋅

ln 𝑎

with 𝑒 ≔ 2.71828… Euler number

Examples

Find the derivatives of

a) 𝑓 𝑥 = 𝑒−𝑥2

b1) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the product rule)

b2) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the quotient rule)

c) ℎ 𝑥 = log𝑎 2 𝑥

Rules of differentiation: Overview (Formulary)

Mathematics 2 for Business Schools Section 1: Fundamentals ...

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