05 2016 - 2017 H2 Maths Differentiation and its Applications Notes (Final)

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 1 of 32 National Junior College 2016 – 2017 H2 Mathematics Topic 5: Differentiation and its Applications Key Questions to Answer: 1. What does the derivative tell you about the behaviour of the graph ? - What does differentiation by first principles mean? - How do you differentiate polynomials by first principles? - What is the geometrical interpretation of f ()x ? 2. What are the meanings of d d y x and d d )f (x x ? - Is d d y x a fraction? - Does d d x y exist? If it does, what does d d x y mean? 3. How do you differentiate the following functions - polynomial functions, - trigonometric functions, - exponential functions, - logarithmic functions, - as well as constant multiples, sums and differences of any combination of the above functions? 4. What are the rules that are useful in differentiation and how do you use them? 5. Under what circumstances would you need to use implicit differentiation, parametric differentiation or logarithmic differentiation? 6. What does 2 2 d d y x mean? - Is this the same as dd dd y xx   or 2 d d y x   ? 7. How do you relate the concavity of the graph with the first and second derivative? - Is the graph of 1y x an example of a graph that is concave upwards? Why or why not? 8. How do you identify a local maximum or minimum on the graph? - What is the difference between a local and a global maximum/minimum? - What is the first derivative test? - What is the second derivative test? Why does it work? Does it work all the time? - When should you use the first derivative test instead of the second derivative test? 9. How do you find the equation of the tangent? - What is the relationship between the gradient of the tangent and that of the normal? - How do you find the equation of the normal? 10. What do you see in common between chain rule, implicit differentiation, parametric differentiation and connected rates of change? Note: This topic is a build -up from the basic ca lculus knowledge acquired under the O -level Additional Mathematics syllabus. Knowledge of O-level calculus is assumed.

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 2 of 32 §1 Differentiation Techniques 1.1 Differentiation by First Principles Recall that the gradient of a straight line is the rate of change of y with respect to x, i.e. change in .change in y x For example, the line 2yx has gradient 2, which means that every unit change in x results in 2 units change in y or equivalently, the rate of change of y with respect to x is 2. For a curve, recall that the gradient is found by differentiation. For example, for the curve 2,yx d 2.d y xx  Hence the gradients of the curve 2yx at x = – 2 and x = 1 are – 4 and 2 respectively. We can see that the gradient of a curve differs for different points on the curve. How do we find d d y x in the first place? Recall that the gradient of a curve at a given point is equal to the gradient of the tangent to the curve at the same point, as seen from the graph below: To find the gradient of the tangent to the curve at a general point P(x, f(x)) on the curve 2,yx consider the following diagram (left): 1 1 d Gradient of 2.d x y lx   2 2 d Gradient of 4.d x y lx   

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 3 of 32 Gradient of the chord 22δ ( δ ) .δ ( δ ) y x x xPQ x x x x   As Q  P along the curve, i.e. δ0x the chord PQ  tangent at P (see above diagram (right))  gradient of the chord PQ  gradient of tangent at P. Therefore the gradient of tangent to the curve at P is given by δ0 22 δ0 2 2 2 δ0 δ0 d δlimd δ ( δ)lim ( δ) 2 δ (δ )lim δ lim(2 δ) 2. x x x x yy xx x x x x x x x x x x x x xx x              In general, we have the following definition. Definition 1.1.1 (Derivative of a Function) If f is a function, the gradient of the curve f ( )yx is defined to be δ0 d f ( δ ) f ( )f ( ) limd δx y x x xxxx   if this limit exists. We call this limit the derivative or the gradient function of f ( ).x The process of obtaining the derivative by finding δ0 f( δ ) f ( )lim δx x x x x  is known as differentiation by first principles.

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 4 of 32 Example 1.1.2 Find the gradient function of the following curves from first principles. (i) 1 ,y x (ii) sin ,yx (iii) e.xy Solution: (i) Using first principles,           δ 0 δ 0 δ0 δ0 δ0 2 11 f( δ ) f ( ) δf ( ) lim lim δδ δlim δδ δlim δδ 1lim δ 1 . xx x x x x x x x x xx xx x x x x x x x x x x x x x x x x                         (ii) Using first principles,   δ 0 0 δ0 δ 0 δ 0 sin δ sinf( δ ) f ( )f ( ) lim lim δδ 2 δδ2cos sin 22lim δ δsinδ 2lim cos lim δ2 2 cos . xx x xx x x xx x xx xx x x x x x xx x x                             (iii) Using first principles,     δ δ 0 δ 0 δ δ0 δ δ0 f( δ ) f ( ) e ef ( ) lim lim δδ e e 1 lim δ e1 e lim δ e. xx x xx xx x x x x x x x xx xx x x              Note: δ0 sin δlim 1.δx x x  Note: δ0 elim 1.δ 1 x x x   

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 5 of 32 1.2 Differentiation of Functions using Standard Results The following basic derivatives and differen tiation techniques are assumed knowledge from the ‘O’-level syllabus: Basic Functions y d d y x Polynomials ( n  ) nx 1nnx    n ax b   1n an ax b   Trigonometric Functions  sin f ( )x  f ( ) cos f ( )xx   cos f ( )x  f ( ) sin f ( )xx  tan f ( )x   2f ( ) sec f ( )xx   cosec f ( )x    f ( ) cosec f ( ) cot f ( )x x x    sec f ( )x    f ( ) sec f ( ) tan f ( )x x x   cot f ( )x   2f ( ) cosec f ( )xx Logarithmic and Exponential Functions  ln f ( )x   f ( ) f ( ) x x  f ( )e x f ( )f ( ) e xx  Chain Rule If f ( )yu and f ( )ux , then d d d .d d d y y u x u x Product Rule If y = uv where u, v are functions of x, then d d d .d d d y u vvux x x Quotient Rule If uy v where u, v are functions of x and v is non-zero, then 2 dd d dd .d uvvuy xx xv   Refer to Appendix B for ways to use your graphing calculator to  evaluate the derivative at a point, and  plot the graph of the derivative function without doing the actual differentiation.

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 6 of 32 The following are additional results for the A-levels. For proofs, refer to Appendix A: y d d y x Inverse Trigonometric Functions 1sin x 2 1 1 x , 1x    1sin f ( ) x   2 f ( ) 1 f ( ) x x   ,  f1 x  1cos x 2 1 1 x   , 1x    1cos f ( ) x   2 f ( ) 1 f ( ) x x   ,  f1 x  1tan x 2 1 1 x   1tan f ( ) x   2 f ( ) 1 f ( ) x x   Logarithmic and Exponential Functions xa lnxaa f ( )xa f ( )f ( ) ln xx a a   log f ( )a x f ( ) log ef ( ) a x x   Relationship between d d y x and d d x y d1 dd d y xx y  . However, d1 dd d n nn n y xx y  in general. Example 1.2.1 Find d d y x if (a)   1sin 1 2 ,yx  (b) 1tan 1 , 2 xy    (