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Misc 5 - Find derivative: ax + b / cx + d - Chapter 13 Class 11 - Miscellaneous

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  1. Chapter 13 Class 11 Limits and Derivatives
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Misc 5 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): š‘Žš‘„ + š‘ļ·®š‘š‘„ + š‘‘ļ·Æ Let f(x) = š‘Žš‘„ + š‘ļ·®š‘š‘„ + š‘‘ļ·Æ Let u = ax + b & v = cx + d āˆ“ f(x) = š‘¢ļ·®š‘£ļ·Æ So, fā€™(x) = š‘¢ļ·®š‘£ļ·Æļ·Æļ·®ā€²ļ·Æ fā€™(x) = š‘¢ļ·®ā€²ļ·Æš‘£ āˆ’ š‘£ļ·®ā€²ļ·Æš‘¢ļ·® š‘£ļ·®2ļ·Æļ·Æ Finding u & v u = ax + b uā€™ = a + 0 = a v = cx + d vā€™ = c + 0 = c fā€™(x) = š‘¢ļ·®š‘£ļ·Æļ·Æļ·®ā€²ļ·Æ = š‘¢ļ·®ā€²ļ·Æš‘£ āˆ’ š‘£ļ·®ā€²ļ·Æš‘¢ļ·® š‘£ļ·®2ļ·Æļ·Æ = š‘Ž š‘š‘„ + š‘‘ļ·Æ āˆ’ š‘ (š‘Žš‘„ + š‘)ļ·® š‘x + dļ·Æ2ļ·Æ = acx + ad āˆ’ š‘Žš‘š‘„ āˆ’ š‘š‘ļ·® cx + š‘‘ļ·Æ2ļ·Æ = acx + ad āˆ’ š‘Žš‘š‘„ āˆ’ š‘š‘ļ·® cx + š‘‘ļ·Æ2ļ·Æ = ad āˆ’ cbļ·® cx + š‘‘ļ·Æ2ļ·Æ Hence, fā€™ (x) = ššš āˆ’ š’„š’ƒļ·® šœš± + š’…ļ·ÆšŸļ·Æ

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