Monday, November 8, 2021

Derivatives with Surprisingly Imaginary Results

Derivatives with Surprisingly Imaginary Results



Here are three derivatives of functions where complex numbers are involved with further algebraic simplification.  




d/dx √(a - x)^(1/2)



d/dx √(a- x)^(1/2)


= 1/2 ∙ (a - x)^(-1/2) ∙ -1


= -1/2 ∙ 1 ÷ (√(a - x))


Going a step further...


= -1/2 ∙ 1 ÷ (√(-1) ∙ √(x - a))


With √(-1) = i ,  1/i = -i


= i ÷ (2 ∙ √(x - a))



d/dx  arcsin(x + a)



d/dx arcsin(x + a)


= 1 ÷ √(1 - (x + a)^2)


= 1 ÷ √(1 - (x^2 + 2 ∙ a ∙ x + a^2))


= 1 ÷ √(-x^2 - 2 ∙ a ∙ x + 1 - a^2)


Factoring out -1 in the denominator: 


= 1 ÷ √((-1) ∙ (x^2 + 2 ∙ a ∙ x - 1 + a^2))


= 1 ÷ (i ∙ √(x^2 + 2 ∙ a ∙ x - 1 + a^2))


= -i ÷ √(x^2 + 2 ∙ a ∙ x - 1 + a^2)



d/dx e^(√(a - x)) 



d/dx e^(√(a - x)) 


= e^(√(a - x)) ∙ d/dx √(a - x)


= -e^(√(a - x)) ÷ (2 ∙ √(a - x))


With:  √(a - x) = i ∙ √(x - a) and e^(i ∙ Θ) = cos Θ + i ∙ sin Θ


= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))


= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))


=  i ∙ e^(i ∙ √(x - a)) ÷ (2 ∙ √(x - a))


= (i ∙ (cos √(x - a) + i ∙ sin √(x - a)) ÷ (2 ∙ √(x - a))


= (-sin √(x - a) + i ∙cos √(x - a)) ÷ (2 ∙ √(x - a))




Eddie


All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


1 comment:

  1. I am happy to visit your blog, a lot of things I can take the benefits of each of your articles. thank you
    wordpress
    ufa88kh.blogspot
    youtube
    ខ្លា នាគ Online

    ReplyDelete

TI-84 Plus CE and CE Python 5.7 update

 TI-84 Plus CE and CE Python 5.7 update Texas Instruments has released software updates, 5.7, for both the TI-84 Plus CE and TI-84 Plus CE P...