Derivatives and Differentiation

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Basic Rules of Differentiation

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Derivatives : 

1 Derivatives This is a revision lesson on derivatives. It is very important your go through your course notes before attempting these review lessons. This revision lesson will cover the basic rules of differentiation. They are designed to help you understand better. The course notes cover comprehensively all topics that could be tested in the examinations.

Derivatives & Differentiation : 

2 Derivatives & Differentiation In the previous lessons, you had learnt about functions Functions can represent sales, cost, profit, distance, population size, etc. In business, often we want to know the rate of change or the slope of the function Slope of a distance function is the speed or velocity function Slope of sales (revenue) function is the marginal revenue function Slope of cost function is the marginal cost function Slope of the profit function is the marginal profit function

Why Derivatives or Differentiation? : 

3 Why Derivatives or Differentiation? Take the slope of the profit function If we have a positive slope, by increasing output we could add profit If we have a negative slope, we can increase profit by reducing output (the opposite) Take the slope of the distance function For a train system, the slope at different points of the track indicate the speed of the train. Is this speed safe and optimum for the train?

Basic Rules of Differentiation : 

4 Basic Rules of Differentiation We use the notation to mean the derivative of f with respect to x at x. These rules could be used in combination

Basic Rules of Differentiation : 

5 Basic Rules of Differentiation Rule 1: Derivative of a Constant The derivative of a constant function is equal to zero e.g. If f(x)=28, then f’(x)= (28)=0

Basic Rules of Differentiation : 

6 Basic Rules of Differentiation Rule 2: The Power Rule If n is any real number, then If f(x)=x, then f’(x)= (x)=1.x1-1=x0=1 If f(x)=x3, then f’(x)= (x3)=3.x3-1=3x2 If f(x)=x5/2, then f’(x)= (x5/2)=(5/2)x3/2

Slide 7: 

7 Basic Rules of Differentiation Rule 3: Derivative of a Constant Multiple of a Function The derivative of a constant times a differentiable function is equal to the constant times the derivative of the function Example f(x) = 5x2, Let g(x) = x2, where g is also function then f(x) = 5g(x) & g’(x) =2x f’(x) =5g’(x) =5 x 2x = 10x

Slide 8: 

8 Basic Rules of Differentiation Rule 4: The Sum Rule The derivative of the sum (difference) of two differentiable functions is equal to the sum (difference) of their derivatives Example f(x) =5x2 + x + 3, Let v(x) = 5x2, u(x) = x, z(x)=3. v, u & z are also functions f(x) =v(x) + u(x) + z(x) f’(x) =v’(x) + u’(x) + z’(x) f’(x) = 10x + 1 + 0 = 10x + 1

Using Basic Differentiation rules : 

9 Using Basic Differentiation rules Now we can apply the combination of differentiation rules to solve problems on derivatives. Example 1. Derivative of Constant Rule, 2 The Power Rule, 3 Derivative of Constant Multiplier of a function rule & 4 Sum Rule

Using Basic Differentiation Rules : 

10 Using Basic Differentiation Rules Another example 1. Derivative of Constant Rule, 2 The Power Rule, 3 Derivative of Constant Multiplier of a function rule & 4 Sum Rule

End of Lesson : 

11 End of Lesson Review Slope or rate of a function is the derivative of the function e.g. f(x), derivative f’(x) = f(x) 4 Basic Rules of Differentiation Derivative of a constant Rule The Power Rule Derivative of Constant Multiple of a function Rule The SUM Rule