Higher Mathematics for Science, Technology and Engineering

Derivatives and Their Applications

Author(s): S. G. Ahmed

Pp: 52-108 (57)

DOI: 10.2174/9781681082813116010003

* (Excluding Mailing and Handling)

Abstract

The basis of the derivative for functions of one variable is introduced herein. The chapter started from the basic definition of the derivative, geometric meaning and growing up to the derivatives of the basic functions. The rules of the derivatives are presented in some details so as the reader can be easily familiar with the different rules of derivatives.


Keywords: Derivatives, domain, elementary basic functions, range.

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1518
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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      680μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 2002 and `b`.`banner_from_date` <= 1740527576 and `b`.`banner_to_date` >= 1740527576 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.1msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 2002
      • 3: 1740527576
      • 4: 1740527576
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='168b8493f79ebdfb8db1743028ff5f20'
      240μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    4Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        DG7GPlCdvytUkhPkhN7HczpuPv2jcdISMmGvl2YM
        uc_cart_id
        168b8493f79ebdfb8db1743028ff5f20
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9326" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/9326
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6Ikx5YjluMDU1QWlIQmFYSEFmOGcxWXc9PSIsInZhbHVlIjoiamliVU9MYTg2cUZkZFhIQ3dNMlp5MTBJY1ZpSmhzMWRwSHhPbmM0ZE0yOD0iLCJtYWMiOiJhN2I4MjJiMjA1NmYxYjZlYWI4MzhjMjllMmQwOTMxMWRiZjNmYjU0OWRlM2E5ODg0ZDJlZWJiNDYxNGE2ZTMyIiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6Ikx5YjluMDU1QWlIQmFYSEFmOGcxWXc9PSIsInZhbHVlIjoiamliVU9MYTg2cUZkZFhIQ3dNMlp5MTBJY1ZpSmhzMWRwSHhPbmM0ZE0yOD0iLCJtYWMiOiJhN2I4MjJiMjA1NmYx" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 25 Feb 2025 23:52:56 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IlY2TkZrTjVhVmVacmY0eWd2ZThEdnc9PSIsInZhbHVlIjoiekFZaGltRCtnS051Nk4vbG1nRlBuWDdVV0NWOEViZnZMZHFHc2NzMi93NzRiUlJHQWE4cmlyS1lWQVE3b3hFWjYxYmwxNjBOQmpDMjQ2bUg0dGhYaElzZWZ4OSswcTUyQkd6YVdHWXBMUXM2V0wrQTAzUkZCY2NoYklxVWN2UFoiLCJtYWMiOiJjYjY0YmNlOTg5NGE0ZmNjZjBhODA3M2IzODQ0NmU3ZTE1ZGM0MDBlOGYwOWUzMWQ0NjNkMTYyN2JmMjUzNzdmIiwidGFnIjoiIn0%3D; expires=Fri, 28 Feb 2025 11:52:56 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6IlY2TkZrTjVhVmVacmY0eWd2ZThEdnc9PSIsInZhbHVlIjoiekFZaGltRCtnS051Nk4vbG1nRlBuWDdVV0NWOEViZnZMZHFHc2NzMi93NzRiUlJHQWE4cmlyS1lWQVE3b3hFWjYxYmwxN" 1 => "alphaeurekaselectnet_session=eyJpdiI6ImVSdlFLU0t5VFVQODhKQ1RGUmJRUmc9PSIsInZhbHVlIjoic1dQRitSRVQ3YnNuN1dTbU1hK0MzTWdDZHkza2cyKzVRZjkvNURhd1BOYlpzWW9kRVdsTEszcUNBWXUzaWxyTnVXUGFvTFBuUDNXek01cERrY3BmOEhwUlZLK1NlMXFVWnMrUEdDUy9xdkpGbElCeDVNR3pKNy82TkNoemxnRVIiLCJtYWMiOiJmMzQ4MzJlOTU4NmEwNDg5NjQ2NTRmOGY1ZWZhNjMxZDk1OTJmN2QyZmE0ODMyM2ViYTc2YjdkNWJjNDlhMjViIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6ImVSdlFLU0t5VFVQODhKQ1RGUmJRUmc9PSIsInZhbHVlIjoic1dQRitSRVQ3YnNuN1dTbU1hK0MzTWdDZHkza2cyKzVRZjkvNURhd1BOYlpzWW9kRVdsTEszcUN" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IlY2TkZrTjVhVmVacmY0eWd2ZThEdnc9PSIsInZhbHVlIjoiekFZaGltRCtnS051Nk4vbG1nRlBuWDdVV0NWOEViZnZMZHFHc2NzMi93NzRiUlJHQWE4cmlyS1lWQVE3b3hFWjYxYmwxNjBOQmpDMjQ2bUg0dGhYaElzZWZ4OSswcTUyQkd6YVdHWXBMUXM2V0wrQTAzUkZCY2NoYklxVWN2UFoiLCJtYWMiOiJjYjY0YmNlOTg5NGE0ZmNjZjBhODA3M2IzODQ0NmU3ZTE1ZGM0MDBlOGYwOWUzMWQ0NjNkMTYyN2JmMjUzNzdmIiwidGFnIjoiIn0%3D; expires=Fri, 28-Feb-2025 11:52:56 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6IlY2TkZrTjVhVmVacmY0eWd2ZThEdnc9PSIsInZhbHVlIjoiekFZaGltRCtnS051Nk4vbG1nRlBuWDdVV0NWOEViZnZMZHFHc2NzMi93NzRiUlJHQWE4cmlyS1lWQVE3b3hFWjYxYmwxN" 1 => "alphaeurekaselectnet_session=eyJpdiI6ImVSdlFLU0t5VFVQODhKQ1RGUmJRUmc9PSIsInZhbHVlIjoic1dQRitSRVQ3YnNuN1dTbU1hK0MzTWdDZHkza2cyKzVRZjkvNURhd1BOYlpzWW9kRVdsTEszcUNBWXUzaWxyTnVXUGFvTFBuUDNXek01cERrY3BmOEhwUlZLK1NlMXFVWnMrUEdDUy9xdkpGbElCeDVNR3pKNy82TkNoemxnRVIiLCJtYWMiOiJmMzQ4MzJlOTU4NmEwNDg5NjQ2NTRmOGY1ZWZhNjMxZDk1OTJmN2QyZmE0ODMyM2ViYTc2YjdkNWJjNDlhMjViIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6ImVSdlFLU0t5VFVQODhKQ1RGUmJRUmc9PSIsInZhbHVlIjoic1dQRitSRVQ3YnNuN1dTbU1hK0MzTWdDZHkza2cyKzVRZjkvNURhd1BOYlpzWW9kRVdsTEszcUN" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "DG7GPlCdvytUkhPkhN7HczpuPv2jcdISMmGvl2YM" "uc_cart_id" => "168b8493f79ebdfb8db1743028ff5f20" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9326" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 23:52:56GET/chapter/9326112186