STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method [optional] Programme F11: Differentiation

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph Programme F11: Differentiation 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

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph Programme F11: Differentiation The gradient of the sloping straight line in the figure is given as:

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text The gradient of a curve at a given point Programme F11: Differentiation The gradient of a curve between two points will depend on the points chosen:

STROUD Worked examples and exercises are in the text 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:

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function (Second derivatives –MOVED to a later set of slides) Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation The gradient of the chord PQ is and the gradient of the tangent at P is

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Two curves Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Programme F11: Differentiation (a)

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines (b) Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (a) so

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (b) so

STROUD Worked examples and exercises are in the text Derivatives of powers of x A clear pattern is emerging:

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve At Q: So As Therefore called the derivative of y with respect to x.

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of polynomials Programme F11: Differentiation To differentiate a polynomial, we differentiate each term in turn:

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives – an alternative notation Programme F11: Differentiation The double statement: can be written as:

STROUD Worked examples and exercises are in the text Towards derivatives of trigonometric functions (JAB) Limiting value of is 1 I showed this in an earlier lecture by a rough argument. Following slide includes most of a rigorous argument. Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Programme F11: Differentiation Area of triangle POA is: Area of sector POA is: Area of triangle POT is: Therefore: That is ((using fact that the cosine tends to 1 -- JAB)):

STROUD Worked examples and exercises are in the text Derivatives of trigonometric functions and … Programme F11: Differentiation The table of standard derivatives can be extended to include trigonometric and the exponential functions: ((JAB:)) The trig cases use the identities for finding sine and cosine of the sum of two angles, and an approximation I gave earlier for the cosine of a small angle. (Shown in class).

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of products of functions Programme F11: Differentiation Given the product of functions of x: then: This is called the product rule.

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of a quotient of two functions Programme F11: Differentiation Given the quotient of functions of x: then: This is called the quotient rule.

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text 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

STROUD Worked examples and exercises are in the text Functions of a function Differentiation of a function of a function Programme F11: Differentiation Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives:

STROUD Worked examples and exercises are in the text 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 Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method [optional] Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Newton-Raphson iterative method [OPTIONAL] Tabular display of results Programme F11: Differentiation Given that x 0 is an approximate solution to the equation f(x) = 0 then a better solution is given as x 1, 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.