Comprehensive Guide to Differentiation: Methods, Formulas, and Applications

1. C2 Methods of Differentiation

2. Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit • provided this limit exists. Recall

3. Section 1. Fundamental Formulas for Differentiation • Formula 1.1 The derivative of a constant is 0. • Formula 1.2 The derivative of the identity function f(x)=x is the constant function f'(x)=1. • Formula 1.3 If f and g are differentiable functions, then (f±g)'(x)= f'(x)±g'(x)

4. Corollary 1.4 (u1+u2+…+un)’= u1’+u2’+…+un’ • Formula 1.5 (The product rule) (fg)'(x) = f(x) g'(x) + g(x) f'(x) • Corollary 1.6 (u1×u2×…×un)’ = u2×…×un×u1’+ u1u3×…×un×u2’+ u1u2u4×…×un×u3’ +…+ u1×u2×u3×…×un-1×un’ • Corollary 1.7 (cu)’ = cu’ • Formula 1.8

5. 2. Rules for Differentiation of Composite Functions and Inverse Functions • Formula 2.1 (The Chain Rule) Let F be the composition of two differentiable functions f and g; F(x) = f(g(x)). Then F is differentiable and F'(x) = f'(g(x)) g'(x) Proof: Exercise

6. Formula 2.2 • (Power Rule) For any rational number n, • where u is a differentiable function of x and u(x)≠0.

7. Corollary 2.3 For any rational number n, if f(x)=xn where n is a positive integer, then f'(x)= nxn-1

8. Formula 2.4 • If y is differentiable function of x given by y=f(x), and if x=f –1(y) with f’(x) ≠0, then • Practice

9. Section 3 The Number e • A man has borrow a amount of $P from a loan shark for a year. The annual interest rate is 100%. Find the total amount after one year if the loan is compounded : • (a) yearly; (b) half-yearly • (c) quarterly (d) monthly; • (e) daily; (f) hourly; • (g) minutely; (h) secondly. • (h) Rank them in ascending order. • (i) Will the amount increase indefinitely? AnswersGraphs

10. e= = 2.718281828459045… • Furthermore, it can be shown (in Chapter 7 and 8) that: • (1) • (2)

11. Section 4 Differentiation of Logarithmic and Exponential Functions • Define y = ex and lnx = logex.

12. Differentiation of Logarithmic function f(x) = lnx Proof: Proof: By Chain Rule and Formula 4.1

15. Differentiation of Logarithmic and Exponential Functions • Exercises on • Product Rule • Quotient Rule • Chain Rule

16. Logarithmic Differentiation Examples Read Examples 4.2- 4.4

17. Formula 4.4

18. Formula 4.5 Quiz

19. Section 5Differentiation of Trigonometric Function Proof of Formula

20. Graphs of trigonometric functions

21. Section 6 The Inverse Trigonometric Functions

22. y=cosx and y=arccosx

23. y=tanx and y=arctanx

24. y=cotx and y=arccotx y=secx and y=arcsecx y=cscx and y=arccscx Graphs

25. Section 7Differentiation of Inverse of Trigonometric Function Proof of Formula

26. Section 10 Indeterminate Forms and L’Hospital Rule Indeterminate Forms

27. (i) Evaluate limx→a f(x)/g(x) where f(a)=g(a)=0. 1. Evaluate limx→o sin3x/sin2x. L’Hospital: limx→o sin3x/sin2x = limx→o 3cos3x/2cos2x = 3/2 2. limx→o (x-sinx)/x3=limx→o (1-cosx)/3x2 = limx→o(sinx)/6x = limx→o(cosx)/6 = 1/6 How? Why?

29. Proof of 0/0 limx→af(x)/g(x) = limx→a(f(x) – f(a))/(g(x) – g(a)) = limx→a(f(x) – f(a))/(x-a)/(g(x) – g(a))/(x-a) = (limx→a(f(x) – f(a))/(x-a))/( limx→a (g(x) – g(a))/(x-a)) = f’(a)/g’(a)

30. Differentiation of exponential function f(x) = ex • Theorem. Let f(x)=bx be the exponential function. Then the derivative of f is f'(x) = bx f'(0) • Proof • Hope: e is the real number such that the slope of the tangent line to the graph of the exponential function y=ex at x=0 is 1. • Formula 4.3 Let f(x)=ex be the exponential function. Then the derivative of f is f'(x) = ex