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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

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## PROGRAMME F11

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**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**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 ofy 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

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