05_2016_-_2017_H2_Maths_Differentiation_and_its_Applications_Lecture_Questions_(student)_Final

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 1 of 9 National Junior College 2016 – 2017 H2 Mathematics Differentiation and its Applications Lecture Questions Part 1. Key Questions to answer:  How do you differentiate the following functions?  polynomial functions  trigonometric functions  exponential functions  logarithmic functions  constant multiples, sums and differences of any combination of the above functions  How do you use implicit differentiation, parametric differentiation and logarithmic differentiation? Prerequisite knowledge: Differentiation by product rule, quotient rule and chain rule. Lecture Readings: Section 1 Question 1.1 Differentiate the following with respect to x and simplify your answers. (a) 2ln ,1 x x    (b)  2log 1 x . Solution: (a)       2 2 2 2 2 2 22 dd ln ln ln(1 )d 1 d 11 (2 )1 1 2 1 11 x xxx x x xxx x x x x x x x             (b)       2 ln 1ddlog 1d d ln 2 1d ln 1ln 2 d 11 ln 2 1 xxxx xx x    

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 2 of 9 Question 1.2 (Implicit Differentiation) The equation of a curve is given by 3 2 34 3 2x x y y   . Find d d y x in terms of x and y. Solution:   3 2 3 2 2 2 2 2 2 2 22 4 3 2 Differentiating implicitly w.r.t. : dd12 6 3 3 dd d3 3 12 6        d d 4 2             d x x y y x yyx xy x y xx yx y x xy x y x xy x y x                    Question 1.3 (Logarithmic Differentiation) Differentiate (a) tan3 x (b) sin xx with respect to x, leaving your answer in terms of x. Solution: (a) Method 1:    tan 2 tand 3 sec ln 3 3d xx xx  Method 2: Let tan3 xy Taking “ln” on both sides, we get  ln tan ln3yx        2 2 2 tan 1d ln 3 secd d ln 3 secd d sec ln 3 3d x y xyx y yxx y xx    (b) Let sin xyx . Taking “ln” on both sides, we get ln (sin )lny x x . Differentiate implicitly w.r.t. x : 1 d 1 (sin ) (ln )cosd d1 sin (ln )cosd y x x xy x x y y x x xxx     sin 1 sin (ln )cos .xx x x xx  

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 3 of 9 Question 1.4 (Differentiating Inverse Trigonometric Functions) The equation of a curve is given by   1cos 2yx  . Find d d y x in terms of x. Solution:   1 2 cos 2 cos 2 dsin 2 d d 2 2 d sin 14 y x y x yy x y xy x        Question 1.5 (Parametric Differentiation) The curve C is defined parametrically by the equations ln , 1x t t y t    , where 0t . Using parametric