PROGRAMME F11 DIFFERENTIATION

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

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

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

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

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

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:

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:

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

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

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

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

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

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

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

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

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

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

Derivatives of powers of x A clear pattern is emerging:

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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.

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:

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

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.

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