1. PROGRAMME F11
DIFFERENTIATION

2. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

3. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

4. the horizontal distance between P and Q
Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of the sloping line straight line in the figure is defined as:
the vertical distance the line rises and falls between the two points P and Q
the horizontal distance between P and Q

5. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of the sloping straight line in the figure is given as:

6. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

7. Programme F11: Differentiation
The gradient of a curve at a given point
The gradient of a curve between two points will depend on the points chosen:

8. The gradient of a curve at a given point
The gradient of a curve at a point P is defined to be the gradient of the tangent at that point:

9. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

10. Programme F11: Differentiation
Algebraic determination of the gradient of a curve
The gradient of the chord PQ is and the gradient of the tangent at P is

11. Programme F11: Differentiation
Algebraic determination of the gradient of a curve
As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.
For example, consider the graph of

12. Algebraic determination of the gradient of a curve
At Q: So As
Therefore
called the derivative of y with respect to x.

13. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

14. Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
Two curves

15. Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
(a)

16. Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
(b)

17. Programme F11: Differentiation
Derivatives of powers of x
Two curves
(a) so

18. Programme F11: Differentiation
Derivatives of powers of x
Two curves
(b) so

19. Derivatives of powers of x
A clear pattern is emerging:

20. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

21. Programme F11: Differentiation
Differentiation of polynomials
To differentiate a polynomial, we differentiate each term in turn:

22. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

23. Programme F11: Differentiation
Derivatives – an alternative notation
The double statement:
can be written as:

24. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

25. Programme F11: Differentiation
Second derivatives
Notation
Limiting value of
Standard derivatives

26. Programme F11: Differentiation
Second derivatives
Notation
The derivative of the derivative of y is called the second derivative of y and is written as:
So, if:
then

27. Programme F11: Differentiation
Second derivatives
Limiting value of
Area of triangle POA is:
Area of sector POA is:
Area of triangle POT is:
Therefore:
That is:

28. Programme F11: Differentiation
Second derivatives
Standard derivatives
The table of standard derivatives can be extended to include trigonometric and the exponential functions:

29. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

30. Programme F11: Differentiation
Differentiation of products of functions
Given the product of functions of x:
then:
This is called the product rule.

31. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

32. Programme F11: Differentiation
Differentiation of a quotient of two functions
Given the quotient of functions of x:
then:
This is called the quotient rule.

33. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

34. Programme F11: Differentiation
Functions of a function
Differentiation of a function of a function
To differentiate a function of a function we employ the chain rule.
If y is a function of u which is itself a function of x so that:
Then:
This is called the chain rule.

35. Programme F11: Differentiation
Functions of a function
Differentiation of a function of a function
Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives:

36. Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Second derivatives
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method

37. Programme F11: Differentiation
Newton-Raphson iterative method
Tabular display of results
Given that x0 is an approximate solution to the equation f(x) = 0 then a better solution is given as x1, where:
This gives rise to a series of improving solutions by iteration using:
A tabular display of improving solutions can be produced in a spreadsheet.

38. Programme F11: Differentiation
Learning outcomes
Determine the gradient of a straight-line graph
Evaluate from first principles the gradient of a point on a quadratic curve
Differentiate powers of x and polynomials
Evaluate second derivatives and use tables of standard derivatives
Differentiate products and quotients of expressions
Differentiate using the chain rule for a function of a function
Use the Newton-Raphson method to obtain a numerical solution to an equation