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