1. Differentiation Safdar Alam

2. Table Of Contents Chain Rules Product/Quotient Rules Trig Implicit Logarithmic/Exponential

3. Notations of Differentiation In functions you will see: -f’(x) -y’(x) These symbols are used to tell that the function is a derivative Derivative: lim f ( x + h ) – f ( x ) h> 0 h

4. Formula for Derivative Nu↑ (n-1) N, standing for a constant (which a derivative of a constant is zero) U, standing for a function Example: X₂ Answer: 2x

5. Practice Problems F(x)= 5x₄ F’(x)= F(x)= x₂+3x₂ F’(x)=

6. Work Page

7. Chain Rules Definition: Formula for the derivative of the two function There are two types of chain rules. (Product/Quotient Rule) Product: (F*DS + S*DF) Quotient: (B*DT – T*DB) B ₂

8. Product Rule Used for Multiplication Product: (F*DS + S*DF) (First * Derivative of Second + Second * Derivative of First) Example: Y = (4x + 3)(5x) Y’= (4x + 3)(5) + (5x)(4) Y’= (20x + 15) + (20x) Y’= 40x + 15

9. Practice Problem Y= (6x + 4)₂(25x + 13) Y’=

10. Work Page

11. Quotient Rule Used for Division Quotient: (B*DT – T*DB) B ₂ (Bottom * Derivative of Top – Top * Derivative of Bottom over Bottom Squared) Example F(x) = (5x + 1) x F’(x) = (x)(5) – (5x + 1)(1) F’(x) = (5x) – (5x + 1) x ₂ x ₂ F’(x) = ( -1 ) x ₂

12. Practice Problem F(x)= 2, (4x + 1)₂ F’(x)=

13. Work Page

14. Trig Functions Derivative of Trig. Functions Sin(x) = Cos(x) dx Cos(x) = -Sin(x) dx Sec(x)= (secx)(tanx) dx Tan(x)= Sec ₂ (x) dx Csc(x)= -(cscx)(cotx) dx Cot(x)= -csc ₂ (x) dx Example: Y= cos(x) + sin(x) Y’= -sinx + cosx

15. Practice Problems Y= tanx sinx Y’=

16. Work Page

17. Implicit Differentiation We use implicit, when we can’t solve explicitly for y in terms of x. Example: Y ₂ = 2y dy dx

18. Practice Problems F(x) = x₃ + y₃ = 15 F’(x) =

19. Work Page

20. Logarithmic Differentiation This applies to chain rules and properties of logs Rules of Log Multiplication- Addition Division- Subtraction Exponents- Multiplication Some key functions to remember ln(1) = 0 ln(e) = 1 ln(x) x = xln(x)

21. Practice Problems Y= (3) x Dy/Dx=

22. Work Page

23. Exponential Diff. F’(x) e(u)= e(u) (du/dx) -Copy the Function and take the derivative of the angle Examples: Y = e(5x) Y’= 5e(5x) Y= e(sinx) Y’= e(sinx)*(-cosx)

24. Practice Problems F(x)= e(5x + 1) F’(x) = F(x)= e(tanx) F’(x)=

25. Work Page

26. Derivative of Natural Log Y= ln(x) Y’(x)= 1/u * du/dx Examples Y= ln(5X) Y’= 5/5X = 1/X

27. Practice Problems Y= ln(ex) Y’= Y=ln(tanx)

28. Work Page

29. FRQ 1995 AB 3 -8x₂ + 5xy + y₃ = -149 A.Find dy/dx -16x + 5x(dy/dx) + 5y + 3y₂(dy/dx) = 0 (dy/dx)(5x + 3y₂) = 16x – 5y dy/dx = 16x – 5y 5x + 3y₂

30. FRQ 1971 AB 1 - ln(x₂ - 4) E. Find H’(7) 1 * (2x) 2x. (X₂ - 4) (X₂ - 4) 2(7) 14. 14 ( (7)₂ - 4 ) (49 – 4 ) 45

31. Sources http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.html http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html

32. © Safdar Alam March 4, 2011