Download presentation

Presentation is loading. Please wait.

Published byLenard Lindsey Modified over 6 years ago

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

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google