17-10-2014 (Calculus Lecture 5th semester BBA)

17/10/2014

1)      Power Rule Function: f(x) = {u(x)}ⁿ èf´(x) = n {u(x)}^n-1  . dy/dx u(x)

f(x) = (3x+5)² è f´(x) = 2 (3x+5) ^2-1. 3 è f´(x) = 6(3x+5) è f´(x) = 18x + 30                                Answer = 18x + 30

2)      Exponential Function: f(x) – e^x è f´(x) =e^x . dy/dx (x)

f(x) = e^3x èf(x) = e^3x . dy/dx (3x) èf(x) = e^3x . 3                                                                  Answer = 3e^3x

3)      Natural Log: f(x) = log (x) è f´(x) = (1/x).dy/dx (x)

f(x) = log (5x) èf´(x) =(1/5x). dy/dx (5x) èf´(x) = (1/5x) . 5                                         Answer = 1/x

f(x) = log (4x² – 16x) è f´(x) = {1/(4x² – 16x)} dy/dx (4x² – 16x) f´(x) = {1/(4x² – 16x)} . (8x – 16)

f´(x) = {4(2x – 4)}/{4(x² – 4x)} è f´(x) = (2x – 4)/(x² – 4x)                                 Answer = (2x – 4)/(x² – 4x)

4)      Chain Rule: dy/dx = {(dy/du) .(du/dx)}

y(u)        = 20 – 3u,                                                             u(x)        = 5x – 4

dy/du   = 20 – 3u                                                              dy/du    = 5x – 4

dy/du   = 0 – 3(1)                                                             dy/du    = 5(1) – 0

dy/du   = – 3                                                                      dy/du    = 5

dy/dx = {(dy/du) . (du/dx)} èdy/dx = ( - 3 . 5) èdy/dx = – 15                                    Answer = – 15

Note: page 712 full exercise 15.6