Problem 52120. Compute the fractional derivative

Created by ChrisR

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/1370\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 1370</w:t></w:r></w:hyperlink><w:r><w:t> asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"q\"/></w:customXmlPr><w:r><w:t>q</w:t></w:r></w:customXml><w:r><w:t>th derivative as </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"D^q x^a\"/></w:customXmlPr><w:r><w:t>D^q x^a</w:t></w:r></w:customXml><w:r><w:t>. Then a familiar example from calculus would be </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"D^2 x^3 = 6 x\"/></w:customXmlPr><w:r><w:t>D^2 x^3 = 6 x</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Fractional calculus involves derivatives in which the order </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"q\"/></w:customXmlPr><w:r><w:t>q</w:t></w:r></w:customXml><w:r><w:t> is not an integer. With </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"q = 1/2\"/></w:customXmlPr><w:r><w:t>q = 1/2</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"a = 2\"/></w:customXmlPr><w:r><w:t>a = 2</w:t></w:r></w:customXml><w:r><w:t>, then </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\"/></w:customXmlPr><w:r><w:t>D^{1/2} x^2 = \\frac{8}{3\\sqrt{\\pi}} x^{3/2}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that computes the fractional derivative of order </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"q\"/></w:customXmlPr><w:r><w:t>q</w:t></w:r></w:customXml><w:r><w:t> of an expression of the form</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f(x) = c1 x^a1 + c2 x^a2 + c3 x^a3 +...\"/></w:customXmlPr><w:r><w:t>f(x) = c_1 x^{a_1} + c_2 x^{a_2} + c_3 x^{a_3} + \\dots </w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The first input to the function will be a 2x</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> matrix in which the first row is the coefficients </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"ci\"/></w:customXmlPr><w:r><w:t>c_i</w:t></w:r></w:customXml><w:r><w:t> and the second row is the exponents </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"ai\"/></w:customXmlPr><w:r><w:t>a_i</w:t></w:r></w:customXml><w:r><w:t>. The output should be in a similar form. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 256.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 128.275px; transform-origin: 407px 128.275px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/1370"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Cody Problem 1370</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 7.79167px; transform-origin: 294.442px 7.79167px; unicode-bidi: normal; "><span style=""> asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">q</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.0083px 7.79167px; transform-origin: 49.0083px 7.79167px; unicode-bidi: normal; "><span style="">th derivative as </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="D^q x^a" style="width: 32px; height: 19px;" width="32" height="19"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; "><span style="">. Then a familiar example from calculus would be </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAmCAYAAADnTCg7AAADgUlEQVR4nO2aUbWrMBBFjwccYAADKEABDuqgDmqhGioBD7VQDbVw30c6K4fcBBI6aeG+2WvloxQITCaTMxMAwzAMwzCMfdMB6AG0FfvoP9CH8QYjgCeAH2oX6A5Y2McTwEnx/oYCA4AHXFSQ3zJoV6U++lcfPf2WPixC7Ig7vCMIA/zsbRT6mCL3ubz6GBTu/1/Twq+9a22Nc+K4hHMNZ0j1W/P+n6KFW+4uAG74wtLXws2oO+br/A/cLJyC/26v80sM/wMX2mvxwLE1wwhv4yvyJl5VeswdIQy5DXw4FkfJWaM7eBGpTQPnnLHl6Qg0cHaUyfJ1JxBOmKvzFCOdd8e6Q5yhpxeYAc4R+JmPJCA7eOGbO7E+Bhv2tnLulHluA/fCNYVdCx9ia0SfGohdJCLsLqpx3j6unDtgvqSkXuZTIkiMe/9AXxrwZNpdBiTrurSckMXnx7KIM/RqCzmI0N074TK7OyQ1K1H9YYbBjJFjgHOylHZokC4tS2l7iQnHWCbYbhKBJb0vXS7etVkUDlu5BuVr2MNHxB1KlHN47IK5gbioxGpbQmrMoVosL1cxxJDvthLhF0bgE5yt+NgTziZLk6bUZkUspZQpWGOIM/T0ewpabP+ggTfmle4ng3qHizATHee9CLn+jnWdExKm0ltbiSbijE0GcXw9y4D5IMYqrcA2m2UTisFc+JobfM4fOgG3pfSSB2eE8/6Y5hjpfiJQt6St3cqz5raSmcd1mlQWxgO8FqVzbbbpAcMwnvMQKQG5BY40uxRXb5ITvhs6J6c+o2ozXntyB5UdqDgULcDG2l3urUDu+/F5OcJZxWbshSU3Y9GjOYM5qzn6plMMLuwt2ZrtsKZJ1GzGOe9SCTp1jXbRhAVW7Tr9N7IJHril9ytxBjWbsVjJER4N5lFBs6gUpl21awbfyCZYrC9dl+s0qjbjgc1Jzdh5NLekxckkU9BefmJ8I5sA8r78Ejsvbb6p2qy0BM2OsJYmlnKD9+rYGnjCDmv4G2HxnbIhf9uQQtVm4ZZ16sF6zDMOjfA9wL1AB/fCnNKGubN8+PFX4B3L2GDL+z8wn6DVbNbidxn0hLkwOmPuBFfo7blzvw/8dsTnyv9Hp4O3P385Lh/3xj7WqWIzruAttSucQ9RQ9WfqI/bQw8r/f4URfpbLRltKu5nNDMMwDMMwDMMwDMMwjP3xD7SJ+V95NDbPAAAAAElFTkSuQmCC" alt="D^2 x^3 = 6 x" style="width: 65.5px; height: 19px;" width="65.5" height="19"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; "><span style="">. </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.725px 7.79167px; transform-origin: 179.725px 7.79167px; unicode-bidi: normal; "><span style="">Fractional calculus involves derivatives in which the order </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">q</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.1667px 7.79167px; transform-origin: 71.1667px 7.79167px; unicode-bidi: normal; "><span style=""> is not an integer. With </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAlCAYAAABf0feeAAACl0lEQVRoge2aa5GDMBhFjwccYKAGUFAFOKgDHNRCNSChHmqhGmqB/ZF8Q9rNk6Rb0s2ZyTCzQMjmQnJzU2g0Go1Gw0oPdJ9uRMPOFZg+3YjGb3pg0cdGIQZdcjkBt8A1B33dpEuJ534lA2oYWlAdlsvdU09vPOu13FCiNbB3VK44B9xDWgc8WIW4ooQ0n3+nCUSPEkKOpcQ5A7Pj3IwS57XzB5RY0gbX/f+SgXLi3IHR8ncxCa6vQs4vKAGbDdeUEueIu2Mn1FflYzba8e+HNqGUOBddbIyErfVEIXF6/cCJMg7nk5QQRyb7Y0Y7RJzH1gp61Nux6OOgK72wOhHX22OraypQctcIJcQZUfNNDjKshYY/KweUAA9+d4jpNmI7y+yUnJL75ZYQZ2ZjpxrIy52cLMhkZxMG1vVCitMo9eXkTp654oScWAyjriM5j+tZVXU1Xs7X6NFzxYmJa3x0qCHxygYLLV+FqwGyKl6we/y9kyvObeN9wgUlTrIwR9aGu5zIxbimxiQ2RxxfXBPDiBp1Nt0vDsJl78yV7TWx7m9wa764JkSWMBDOe8xVbepk9g1uzRXXhMgWxmy0zSaOPHdSqlup3a3JkJ86VwyEhZFFvrcSafTrwlIWXaFhrwa2iuOLa1zIWtGXJHQok+FNGzqe9xkG1rdd1jupqcAeMU1PijipcY25iL96yoPItMEM48wdu544J7d3jjynGzfiDIbMGbGIMLHzaPT8fWANOM3xXSx0rfsPobnM9z+lxjUSEseW7P6sORXIQYb73e65lNw9rI0T+Qn0WzlTdyqQQ25c83bkVyO7foPewG5/MNihhjMzFbhTr1Pbwpn0mOpPkF1PW6kxjd7C1rim8QfM1LlsaDQajb3xA5BnUMSw0QAvAAAAAElFTkSuQmCC" alt="q = 1/2" style="width: 51.5px; height: 18.5px;" width="51.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAABqUlEQVRoge2YW5GDMBRAjwccYAADKKgCHOAAB2sBDZWAh7VQDVhoPy53k3Z5JMC0yfSemXy00E44yX0EMAzDMAwjfyqgBbpp1J+dTlqUwADcZ8YvIu+rKYARJ2QAbjyLuvHloq6IpFcJNSJNRV3fPK9kKBEBS7tEr98RkcWb5pUUHfCzcc8VJ+orQ65BdssaHSdJ8kunls16+py7fZU07v2DClcNWkRMz3PC21oppcT1J0fG2b2NhttWWM7S4Mrja0LzS2coNfN9Suxo9zzMCtoehC72H2uCwE04xv5ZO+nM8Nbn7GJ/eMFJmNva5cb1XCiQTTAQWfr97nSpuVL7ufcVPcuRskpIOdQz0LB3dgnQIIscnYfA7aIlAX7yjY3jVKrbIUEhAvyTdGwCTaG6HRIEz6F2mbne4nbannz06epWsy2oREQusiapQZKcNl4ajj07yucHqBBBc4uvFEiTvHYPFU6SlsUCkaBVQHdSP40cXi+ooBF5rqUxEtgc+ydi/z2Lhtb48n3qqKDQfBccFZfp5pb/8ZvbobYhLt/l3PcZhmEYhmEYqfEAiYzgTN2anFYAAAAASUVORK5CYII=" alt="a = 2" style="width: 36.5px; height: 18px;" width="36.5" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.4417px 7.79167px; transform-origin: 19.4417px 7.79167px; unicode-bidi: normal; "><span style="">, then </span></span></div><div style="block-size: 36.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.4583px; text-align: left; transform-origin: 384px 18.4583px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-17px"><img src="data:image/png;base64,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" alt="D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))" style="width: 117px; height: 37px;" width="117" height="37"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 196.292px 7.79167px; transform-origin: 196.292px 7.79167px; unicode-bidi: normal; "><span style="">Write a function that computes the fractional derivative of order </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">q</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.675px 7.79167px; transform-origin: 88.675px 7.79167px; unicode-bidi: normal; "><span style=""> of an expression of the form</span></span></div><div style="block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="f(x) = c1 x^a1 + c2 x^a2 + c3 x^a3 +..." style="width: 204px; height: 26px;" width="204" height="26"></span></div><div style="block-size: 43.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.8167px; text-align: left; transform-origin: 384px 21.8167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; "><span style="">The first input to the function will be a 2x</span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.775px 7.79167px; transform-origin: 143.775px 7.79167px; unicode-bidi: normal; "><span style=""> matrix in which the first row is the coefficients </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAoCAYAAAALz1FrAAAA9ElEQVRIie2VbQ2EMAyGHw9zMAMYOAUoOAc4wAEW0IAEPGABDWdh92NtRgjf2y65ZE/CDxi06du3BQqFQqEQjQVqoJXLpgz+AkZgBt5yPwAOaFIk6CTYsHquSWbAxCToJdC4cTbL2RSToJEgH7a1t3jZHldhJLjDy5UFrcIB1c1vDd559dmLI0Gqu3K8CYY4RKt40lSt5FSBmCSXmTieAYO3d5TrWkI16wZW+J6tJ90S5sqdJVCWze8kcY+vbk/vmu3tcEhFWIZXmqkrKMk+20N7mXQ7L7H8wJE6hNnWEARn6b8mi2TqxoGMjVcn3l2ohcI/8AWjzEUn2YFV5AAAAABJRU5ErkJggg==" alt="ci" style="width: 12.5px; height: 20px;" width="12.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83.625px 7.79167px; transform-origin: 83.625px 7.79167px; unicode-bidi: normal; "><span style=""> and the second row is the exponents </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAAoCAYAAAAPOoFWAAABF0lEQVRYhe2UUQ3DIBCGPw84mAEMoGAK5qAO6mAW0FAJeJiFaqiF7aF3KWNlkJQt2cKX8FD+9n7uehx0Op1O52ewwACMgJM9J8+2pUkAZjFzgAduwF3WqYXRRYLNgEm0e6R91Cg2ux41OkfB3I5+KujVGGCRQFPmHc16YT/raka2U+e6LIgeMrqlsks1q1wgFx1mzLzjRffvjGoCBcqZa2ZvSxyX8LyjD2yZN/1fqdmF9RpMPJfZ81oFS0WX2sgssJ7cSDC9b5qZlxV3rKPcPE/oyeM1sZVsSfZTrqINNWawlnCUD9K5Vxq+OjObDecchobzsoROlsPzsga9zHvXpjmzmBkqu/EI+r9ufKFBtIsPTZZO5894AM2JYzwwHTn5AAAAAElFTkSuQmCC" alt="ai" style="width: 13.5px; height: 20px;" width="13.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; "><span style="">. The output should be in a similar form. </span></span></div></div></div>

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fractional calculus special functions

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Problem 52120. Compute the fractional derivative

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Problem 52120. Compute the fractional derivative

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