Victoria School AM #12 Differentiation 2024

Secondary 4 Additional Mathematics Differentiation 1 Name: Class Reg No. Date: 4 Chapter 12: Differentiation Reference Book: Additional Maths 360 Volume B, Marshall Cavendish You will learn how to, • Relate the derivative of a function to the gradient of tangent. • Differentiate functions using the basic rules of differentiation, ➔ Constant multiple rule: ( ) ( ) dd dd ff xx a x a x =       , where a is a constant ➔ Power rule: d dx x n = nx n – 1 , n is a rational number ➔ Sum and difference rule: ( ) ( ) ( ) ( ) d d d d d d f g f g x x x x x x x =            ➔ Chain rule: d d d d d d y y u x u x=  ( ) 1dd dd nn yu nuxx −= to differentiate functions of the form y = u n where u = f(x) ➔ Product rule: ( )d d d d d d vuuv u vx x x =+ ➔ Quotient rule: 2 dd d dd d uvvuu xx x v v − = Introduction The word "calculus" comes from Latin (calculus) and means a small stone used for counting. Calculus is the mathematical study of change and has two major branches; Differential calculus, which concerns itself with the rate of change and the gradient of curves, as well as integral calculus, which is regarding the accumulation of quantities and area under curves. The First Derivative of y The first derivative of y is denoted by f ( )x or , pronounced as “dee y dee x”. It is the gradient function of y, that is, it is the function that provides the value of the instantaneous gradient of y with respect to x at any required point. ➔ Hence, the gradient (at a particular point) of a curve is defined as the derivative of a function or gradient function. It is the result of differentiating y with respect to x. The process of finding d d y x is called differentiation. Teachers’

Secondary 4 Additional Mathematics Differentiation 2 Notation The notation d d y x , is also known as the first derivative of y with respect to the variable x. If f is a function of x, i.e. f(x), the first derivative of f with respect to x can be denoted by → If y = f(x), then d f ( )d y xx = . Sometimes, you may be asked to find ( )d d nxx . This represents the first derivative of xn with respect to x. ( )d dx is the differential operator. You simply need to differentiate the expression in the brackets with respect to the variable x. Similarly, ( )d dt is the differential operator with respect to the variable t. t is commonly used to represent time. The Power Rule *We “bring down” the power of the variable, followed by reducing the power by one. The Constant Multiple Rule The Constant Rule If nyx= , where n is a rational constant, then 1d d ny nxx −= If ( )f x is a function and a is a constant, then ( ) ( ) dd dd ff xx a x a x =       Given a general function ,yk= where k is a constant, we can use Constant Multiple Rule and the Power Rule to show that d 0.d y x = ( ) ( ) ( ) 0 0 01 dd ()dd d d d