Saturday, April 10, 2021

Tabela e derivateve në matematikë

Funksioni derivues	Funksioni
Funksioni	Funksioni i rregullt¹
$f(x)=0\;$	$F(x)=C\;$
$f(x)=k\;(k\in \mathbb {R} )$	$F(x)=kx+C\;$
$f(x)=1\;$	$F(x)=x+C\;$
$f(x)=x\;$	$F(x)={\frac {1}{2}}x^{2}+C\;$
$f(x)=2x\;$	$F(x)=x^{2}\;$
$f(x)=x^{2}\;$	$F(x)={\frac {1}{3}}x^{3}\;$
$f(x)=3x^{2}\;$	$F(x)=x^{3}\;$
$f(x)=qx^{q-1}\;(q\in \mathbb {R} )$	$F(x)=x^{q}\;$
$f(x)=x^{q}\;$	$F(x)=\left\{{\begin{matrix}{\frac {x^{q+1}}{q+1}},&{\mbox{wenn }}q\neq -1\\\ln \|x\|,&{\mbox{wenn }}q=-1\end{matrix}}\right.$
$f(x)=\sum _{n=0}^{N}nk_{n}x^{n-1}\;$	$F(x)=\sum _{n=0}^{N}k_{n}x^{n}\;$
$f(x)=\sum _{n=0}^{N}k_{n}x^{n}\;$	$F(x)=\sum _{n=0}^{N}{\frac {k_{n}}{n+1}}x^{n+1}\;$
$f(x)=e^{x}\;$	$F(x)=e^{x}\;$
$f(x)=e^{kx}\;$	$F(x)={\frac {1}{k}}e^{kx}\;$
$f(x)=a^{x}\ln a\;(a>0)$	$F(x)=a^{x}\;$
$f(x)=a^{x}\;(a>0)$	$F(x)={\frac {a^{x}}{\ln a}}\;$
$f(x)={\frac {-2}{x^{3}}}\;$	$F(x)={\frac {1}{x^{2}}}\;$
$f(x)={\frac {-1}{x^{2}}}\;$	$F(x)={\frac {1}{x}}\;$
$f(x)={\frac {1}{x}}\;$	$F(x)=\ln \left\|x\right\|\;$
$f(x)=\ln x\;$	$F(x)=x\ln x-x\;$
$f(x)={\frac {1}{x}}{\frac {1}{\ln a}}\;(a>0)$	$F(x)=\log _{a}x\;$
$f(x)=\log _{a}x\;(a>0)$	$F(x)={\frac {1}{\ln a}}(x\ln x-x)\;$
$f(x)=\sin x\;$	$F(x)=-\cos x\;$
$f(x)=\cos x\;$	$F(x)=\sin x\;$
$f(x)=\tan x\;$	$F(x)=-\ln \left\|\cos x\right\|\;$
$f(x)=\cot x\;$	$F(x)=\ln \left\|\sin x\right\|\;$
$f(x)={\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x\;$	$F(x)=\tan x\;$
$f(x)={\frac {-1}{\sin ^{2}x}}=-(1+\cot ^{2}x)\;$	$F(x)=\cot x\;$
$f(x)=\arcsin x\;$	$F(x)=x\;\arcsin x+{\sqrt {1-x^{2}}}\;$
$f(x)=\arccos x\;$	$F(x)=x\arccos \;x-{\sqrt {1-x^{2}}}\;$
$f(x)=\arctan x\;$	$F(x)=x\arctan x-{\frac {1}{2}}\ln \left(1+x^{2}\right)\;$
$f(x)={\frac {1}{\sqrt {1-x^{2}}}}\;$	$F(x)=\arcsin x\;$
$f(x)={\frac {-1}{\sqrt {1-x^{2}}}}\;$	$F(x)=\arccos x\;$
$f(x)={\frac {1}{x^{2}+1}}\;$	$F(x)=\arctan x\;$
$f(x)={\frac {1}{(x^{2}+1)^{2}}}\;$	$F(x)={\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)\;$
$f(x)=\sinh x\;$	$F(x)=\cosh x\;$
$f(x)=\cosh x\;$	$F(x)=\sinh x\;$
$f(x)=\tanh x\;$	$F(x)=\ln \left\|\cosh x\right\|\;$
$f(x)=\coth x\;$	$F(x)=\ln \left\|\sinh x\right\|\;$
$f(x)={\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x\;$	$F(x)=\tanh x\;$
$f(x)={\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x\;$	$F(x)=\coth x\;$
$f(x)=\operatorname {arsinh} \;x\;$	$F(x)=x\;\operatorname {arsinh} \;x-{\sqrt {x^{2}+1}}\;$
$f(x)=\operatorname {arcosh} \;x\;$	$F(x)=x\;\operatorname {arcosh} \;x-{\sqrt {x^{2}-1}}\;$
$f(x)=\operatorname {artanh} \;x\;$	$F(x)=x\;\operatorname {artanh} \;x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}\;$
$f(x)=\operatorname {arcoth} \;x\;$	$F(x)=x\;\operatorname {arcoth} \;x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}\;$
$f(x)={\frac {1}{\sqrt {x^{2}+1}}}\;$	$F(x)=\operatorname {arsinh} \;x\;$
$f(x)={\frac {1}{\sqrt {x^{2}-1}}}\;,\;x>1$	$F(x)=\operatorname {arcosh} \;x\;$
$f(x)={\frac {1}{1-x^{2}}}\;,\;\left\|x\right\|<1$	$F(x)=\operatorname {artanh} \;x\;$
$f(x)={\frac {1}{1-x^{2}}}\;,\;\left\|x\right\|>1$	$F(x)=\operatorname {arcoth} \;x\;$

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brave payments verification

query_vars)) { $id = $wp->query_vars[BRAVE_WELL_KNOWN_URI_QUERY_VAR]; // run the more specific hook first do_action("well_known_uri_{$id}", $wp->query_vars); do_action("well-known-uri", $wp->query_vars); } } } add_filter('query_vars', array('BraveWellKnownUriPlugin', 'query_vars')); add_action('parse_request', array('BraveWellKnownUriPlugin', 'delegate_request'), 99); add_action('init', array('BraveWellKnownUriPlugin', 'add_rewrite_rules')); register_activation_hook(__FILE__, array('BraveWellKnownUriPlugin', 'activate_plugin')); register_deactivation_hook(__FILE__, 'flush_rewrite_rules'); function well_known_uri($query) { $options = get_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME); if (is_array($options)) { foreach($options as $key => $value) { if (strpos($key, BRAVE_WELL_KNOWN_URI_MATCHER_SUFFIX) !== 0) continue; $offset = substr($key, strlen(BRAVE_WELL_KNOWN_URI_MATCHER_SUFFIX) - strlen($key)); $suffix = $options[BRAVE_WELL_KNOWN_URI_MATCHER_SUFFIX . $offset]; if ((empty($suffix)) || (strpos($query[BRAVE_WELL_KNOWN_URI_QUERY_VAR], $suffix) !== 0)) continue; $type = $options[BRAVE_WELL_KNOWN_URI_MATCHER_TYPE . $offset]; if (empty($type)) $type = 'text/plain; charset=' . get_option('blog_charset'); header('Content-Type: ' . $type, TRUE); $contents = $options[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $offset]; if (is_string($contents)) echo($contents); exit; } } status_header(404); header('Content-Type: text/plain; charset=' . get_option('blog_charset'), TRUE); echo 'Not ' . (is_array($options) ? 'Found' : 'configured'); exit; } add_action('well-known-uri', 'well_known_uri'); // (mostly) adapted from Example #2 in https://codex.wordpress.org/Creating_Options_Pages class BraveWellKnownUriSettings { private $options; private $slug = 'well-known-admin'; private $option_group = 'well_known_option_group'; public function __construct() { add_action('admin_menu', array($this, 'add_plugin_page')); add_action('admin_notices', array($this, 'admin_notices')); add_action('admin_init', array($this, 'page_init')); } public function add_plugin_page() { add_options_page('Settings Admin', 'Brave Payments Verification', 'manage_options', $this->slug, array($this, 'create_admin_page')); } public function admin_notices() { settings_errors($this->option_group); } public function create_admin_page() { $this->options = get_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME); ?>

Brave Payments Verification

option_group, BRAVE_WELL_KNOWN_URI_OPTION_NAME, array($this, 'sanitize_field')); $options = get_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME); if (!is_array($options)) $j = 1; else { $newopts = array(); for ($i = 1, $j = 1;; $i++) { if (!isset($options[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) break; if (empty($options[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) continue; /* courtesy of https://stackoverflow.com/questions/619610/whats-the-most-efficient-test-of-whether-a-php-string-ends-with-another-string#2137556 */ $reversed_needle = strrev('_' . $i); foreach($options as $key => $value) { if (stripos(strrev($key), $reversed_needle) !== 0) continue; $newopts[substr($key, 0, 1 + strlen($key) - strlen($reversed_needle)) . $j] = $value; } $j++; } update_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME, $newopts); for ($j = 1;; $j++) if (!isset($newopts[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $j])) break; $j = 1; } for ($i = 1; $i <= $j; $i++) { add_settings_section($section_prefix . $i, 'Enter your Publisher Verification Code Below and click "Save Changes"', array($this, 'print_section_info'), $this->slug); add_settings_field(BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i, $contents_title, array($this, 'field_callback'), $this->slug, $section_prefix . $i, array('id' => BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i, 'type' => 'textarea')); } } public function print_section_info() {} public function field_callback($params) { $id = $params['id']; $type = $params['type']; $value = ''; $prefix = 'options[$id])) $value = esc_attr($this->options[$id]); $suffix = '" />'; } elseif ($type === 'textarea') { $prefix = ''; } echo($prefix . $value . $suffix); } public function sanitize_field($input) { $valid = array(); for ($i = 1;; $i++) { if (!isset($input[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) break; $valid += $this->sanitize_suffix($input, BRAVE_WELL_KNOWN_URI_MATCHER_SUFFIX . $i); $valid += $this->sanitize_type($input, BRAVE_WELL_KNOWN_URI_MATCHER_TYPE . $i); $valid += $this->sanitize_contents($input, BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i); } return $valid; } public function sanitize_suffix($input, $id) { $valid = array(); if (empty($input[$id])) { $input[$id] = 'brave-payments-verification.txt'; $valid[$id] = $input[$id]; return $valid; } if (!isset($input[$id])) return $valid; $result = trim(sanitize_text_field($input[$id]), '/'); if (strstr($result, '/') !== FALSE) { add_settings_error($id, 'invalid_suffix', __('URI path must not contain "/"') . ' - ' . $result, 'error'); return $valid; } $valid[$id] = $result; return $valid; } // a 90% implementation of https://www.w3.org/Protocols/rfc1341/4_Content-Type.html // no self-respecting browser should have problems with a Content-Type header that this considers valid... public function sanitize_type($input, $id) { $valid = array(); $validP = TRUE; if (empty($input[$id])) { $input[$id] = 'text/plain'; $valid[$id] = $input[$id]; return $valid; } $parts = explode(';', $input[$id]); list($type, $subtype) = explode('/', $parts[0]); $token = '/^([0-9A-Za-z' . "'" . preg_quote('!#$%&*+^_`{|}~-') . '])+$/'; $word = '/^([0-9A-Za-z' . preg_quote('!#$%&*+^_`{|}~-') . '])+$/'; $string = '/^"([0-9A-Za-z' . preg_quote('!#$%&*+^_`{|}~-') . ']|(\\"))+"$/'; $type = trim(strtolower(sanitize_text_field($type))); if (empty($type)) { add_settings_error($id, 'missing_mime_type', __('Content-Type missing type'), 'error'); $validP = FALSE; } // skipping "media" types (audio, image, video) if ( (!in_array($type, array('application', 'message', 'multipart', 'text'))) && ((strpos($type, 'x-') !== 0) || (!preg_match($token, $type)))) { add_settings_error($id, 'invalid_mime_type', __('Content-Type has invalid MIME type') . ' - ' . $type, 'error'); $validP = FALSE; } $subtype = trim(sanitize_text_field($subtype)); if (empty($subtype)) { add_settings_error($id, 'missing_mime_subtype', __('Content-Type missing subtype'), 'error'); $validP = FALSE; } if (!preg_match($token, $subtype)) { add_settings_error($id, 'invalid_mime_subtype', __('Content-Type invalid subtype') . ' - ' . $subtype, 'error'); $validP = FALSE; } if (!$validP) return $valid; $result = $type . '/' . $subtype; for ($i = 1; $i < count($parts); $i++) { list($attribute, $value) = explode('=', $parts[$i]); $attribute = trim(sanitize_text_field($attribute)); if (empty($attribute)) { add_settings_error($id, 'missing_attribute', __('Content-Type missing attribute'), 'error'); $validP = FALSE; continue; } if (!preg_match($token, $attribute)) { add_settings_error($id, 'invalid_mime_attribute', __('Content-Type invalid attribute') . ' - ' . $attribute, 'error'); $validP = FALSE; } $value = trim(sanitize_text_field($value)); if (empty($value)) { add_settings_error($id, 'missing_value', __('Content-Type missing value'), 'error'); $validP = FALSE; } if (!(preg_match($word, $value) || preg_match($string, $value))) { add_settings_error($id, 'invalid_mime_value', __('Content-Type invalid value') . ' - ' . $value, 'error'); $validP = FALSE; } $result .= '; ' . $attribute . '=' . $value; } if ($validP) $valid[$id] = $result; return $valid; } public function sanitize_contents($input, $id) { $valid = array(); // nothing to sanitize, it's just raw text if (isset($input[$id])) $valid[$id] = $input[$id]; return $valid; } } if (is_admin()) new BraveWellKnownUriSettings(); ?>

Detyra të zgjidhura nga matematika

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Saturday, April 10, 2021

Tabela e derivateve në matematikë

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