methods of differentiation pdf

London WC1R 4HQ. 75 0 obj<>stream /N 3 I�XPD(��58����%,�T(8"9ph%R`�R*2N�6І "�Ƃ$ 0`�_ck`�)��Ç�=�潇�]��B{�Q� �֣ �GG����[�ت�ci��l����׌zb_ �^ ��\ ��[ �GZ �YJp�PfY�T��F��A��. <]>> endstream endobj 43 0 obj<> endobj 44 0 obj<> endobj 45 0 obj<>stream {{{;�}�#�tp�8_\. Learners faced with such challenges would benefit from 1 - Derivative of a constant function. /Filter /FlateDecode Differentiation requires the teacher to vary their approaches in order to accommodate various learning styles, ability levels and interests. �ӯ���=�Mθ�;�(ZF�� ��J!��J���1�����.��cv��'߃g(;1K���d�� �k�!�P�Qw �1o� Nm� �x�����J��=%^9�U'��9�4Z$��yc���5.� P� g���| �P�f@�70�U0¤�aR��9��nS8��i- ���� C/3;�&O�"&k&��L��Z]YYX�X�޲���p����30 ��@�X8�y�� � � 0000004449 00000 n 3 0 obj endobj h��ZYoI�+��U�N߇Yl�8�C��D�x�f�Ⰰh��U=��/�!N"O��U]�UuW�h� %�����%�XxB���A��'#�kxr�(cP�1�N&4V)¤�PЄi��`��,aj�g�C)��p(0�t7�n8�� �j h�b```a``�a`c`P+bb@ !�;P�cF1��H�H�D�c�Vi(����x�Ȭ"gw�DAY�yu|W8xj����5�4\�� 0000025297 00000 n 3. 1 0 obj 0000025218 00000 n 33 43 0000022814 00000 n Implicit Differentiation Find y if e29 32xy xy y xsin 11 . 0000004019 00000 n h�bbd```b``e�� �q?���̖`��d����ٹ`v���� Rx;�,�i&g��)$f��H�6�g�g��`��7! %���� 641 0 obj <> endobj 685 0 obj <>/Filter/FlateDecode/ID[]/Index[641 113]/Info 640 0 R/Length 165/Prev 733866/Root 642 0 R/Size 754/Type/XRef/W[1 3 1]>>stream Created: May 11, 2017| Updated: Oct 4, 2020, Calculus---Revision-Exercises-on-Methods-of-Differentiation. 0000041887 00000 n 0000007833 00000 n 0000045135 00000 n x���wTS��Ͻ7�P����khRH �H�. Class discussion conducted by teacher (and what else!) �MFk����� t,:��.FW������8���c�1�L&���ӎ9�ƌa��X�:�� �r�bl1� 0000046192 00000 n Square From below The table of values and associated graph above both suggest that as x → −1 from below, f(x) → 1. It is convenient to take the logarithm of the function first & then differentiate. 0000004557 00000 n stream The questions (with separate answers) are contained in seven exercises. 0000010271 00000 n A�ڂ��u��d������;'�ta��ccz~����nߵ���}}��;�]3�k����vٲpM[O�������+K��J��xw��?��,�16ql��{��=��x���n�vM�W������}�����)�����Ct��yIL�7�:Tu���r�^kW���:�]��u3�v��X�����KdeV�g�g�� Rememberyyx here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. � �����Y�H���?�`��'P(��e�@!���@����cXbx��g���� M�Ļ(c��$�� �/E��.d �x��ƕǼ%�w1p�iF �g������3� H��� endstream endobj 642 0 obj <> endobj 643 0 obj <> endobj 644 0 obj <>stream The basic rules of Differentiation of functions in calculus are presented along with several examples . The underlying function itself (which in this cased is the solution of the equation) is unknown. %%EOF 33 0 obj <> endobj Later exercises are more advanced and differentiation may require a combination of methods. /Length 2596 ?���:��0�FB�x$ !���i@ڐ���H���[EE1PL���⢖�V�6��QP��>�U�(j << 5fXL�5���P���*�RH'r�~*���u9�/g8�s���������$8�ԡF;sO�1�Ѣ�ט�h��&�C��z��J}�0��j��!������$S0tuL:l��e!n���'�g0�� trailer 0000014427 00000 n endobj �9�cx��+?L�c�����6M���ٯ�N�p���}0[sm{�/0��{a�1�D|��[��� Integration,unlike differentiation, is more of an art-form than a collection of algorithms. 0000001552 00000 n The derivative of f(x) = c where c is a constant is given by f '(x) = 0 Example f(x) = - 10 , then f '(x) = 0 2 - Derivative of a power function (power rule). %PDF-1.4 %���� !Y��vك�z6s�������IH����@�Qx�x��k���00120�Ch�(9(���� 7�� endstream endobj startxref 0 %%EOF 753 0 obj <>stream 0000010794 00000 n Discussion groups conducted by selected student chairpersons (yes, and what else!) registered in England (Company No 02017289) with its registered office at 26 Red Lion Mosaic Attenuation: Etiology, Methods of Differentiation, and Pitfalls 1 PART A: BASIC RULES OF DIFFERENTIATION In Section 3.2, we discussed Rules 1 through 4 below. This website and its content is subject to our Terms and y˼E�1�_�_�ߘ����������s��=�0r`ȴyؼ12�NO���o�7�/�/ȴy؄�G�x��G�x��G�x��G�x��G�x��G��`Oគ=�{ The constant rule: The derivative of a constant function is 0. Lecture by teacher (and what else can you do!) x�b```b``9�������A��X��,3X���*X�K �4����p��}��,�+��є�nq��kXl[T�)g�`��V��2��]�*��,����Z�i�g�Ut)Pp� F~6۹n�Z�r1�hP{��f���U�@�A����@V1))w A�f���Q�ftqK�@H1(�� q�!f���X2�i ��H0�u(�a��P=��(�����8�u�C���y�&NMF���N�;����[�[li�e��x�юM�a��=�b�DGc ��x�)0�4���w Gze� different ways of learning, different interests, different ways of responding to instruction, and preferred ways of learning or expressing themselves (Ravitch 2007). endstream endobj 41 0 obj[62 0 R] endobj 42 0 obj<>stream Consequently, students are failing because they cannot learn well with the methods through which they are being taught. )*D�Bx�Bz|B{h�#"��!���B ��C ��D ��E Rh�yT����`^�^�P^�0^�)���Ӳ��N�t5�����Mtu|]'��(��v���ї���;Td. 0000044380 00000 n Numerical methods John D. Fenton a pair of modules, Goal Seek and Solver, which obviate the need for much programming and computations. There are certain methods of Tes Global Ltd is �'x?��.Pp�j�m��.Q�����3c 0000000016 00000 n Goal Seek, is easy to use, but it is limited – with it one can solve a single equation, however complicated or however many spreadsheet cells are involved, whether the equation is linear or nonlinear. SECTION 3.3: TECHNIQUES OF DIFFERENTIATION LEARNING OBJECTIVES • Learn how to differentiate using short cuts, including: the Linearity Properties, the Product Rule, the Quotient Rule, and (perhaps) the Reciprocal Rule. 0000044722 00000 n 0000002655 00000 n 0000008647 00000 n 5. �=-���4�F�_3��A� C#�9�Xb3B��2 Y��5�G�#,q�,qx1����+#q%/9q���R� ���S�%P!�H�!�8&��Q��h�\�tPr��( Exercise D involves logarithmic functions and Exercise E is on exponential functions. H�\�_k�@��|�ylJ4s�� �8���^����r���P�� Recitation oral questions by teacher answered orally by students (then what!) 0000009900 00000 n *1 J�� "6DTpDQ��2(���C��"��Q��D�qp�Id�߼y�͛��~k����g�}ֺ ����LX ��X��ň��g`� l �p��B�F�|،l���� ��*�?�� ����Y"1 P������\�8=W�%�Oɘ�4M�0J�"Y�2V�s�,[|��e9�2��s��e���'�9���`���2�&c�tI�@�o�|N6 (��.�sSdl-c�(2�-�y �H�_��/X������Z.$��&\S�������M���07�#�1ؙY�r f��Yym�";�8980m-m�(�]����v�^��D���W~� ��e����mi ]�P����`/ ���u}q�|^R��,g+���\K�k)/����C_|�R����ax�8�t1C^7nfz�D����p�柇��u�$��/�ED˦L L��[���B�@�������ٹ����ЖX�! 0000009440 00000 n endstream endobj 34 0 obj<> endobj 35 0 obj<>/MediaBox[0 0 594 774]/TrimBox[0 0 594 774]/Resources<>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageC]/ExtGState<>>>/Type/Page>> endobj 36 0 obj<> endobj 37 0 obj<> endobj 38 0 obj[/ICCBased 55 0 R] endobj 39 0 obj<> endobj 40 0 obj<>stream )����f�uu���s�k�`G]��O4�����H肕sI-k�8h��. 0000004343 00000 n Techniques of Differentiation - Classwork Taking derivatives is a a process that is vital in calculus. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule). 0000025804 00000 n 0 0000002056 00000 n 0000011779 00000 n >> � I�C,�A�8��K����4,� �Z6�r� �@���R�t C���X��CP�%CBH@�R����f�[�(t� C��Qh�z#0 ��Z�l�`O8�����28.����p|�O×�X 0000004043 00000 n 0000031824 00000 n �CqZ(�!F� �������̰�Fqy�*n�`����o�S�^�O�?9�t��&�-���� �۴v����J��c¡�.��}qL�J_�! Differentiation techniques 151 chapter 4 Solution a To investigate the behaviour of f as x → −1, a table of values or a graph can be used. [/ICCBased 3 0 R] Rules of Differentiation of Functions in Calculus. LOGARITHMIC DIFFERENTIATION : To find the derivative of a function : (a) which is the product or quotient of a number of functions or (b) of the form [f(x)] g (x) where f & g are both derivable. <> In order to take derivatives, there are rules that will make the process simpler than having to use the definition of the derivative. 0000002825 00000 n 150 Teaching Methods 1. ]�b����]+cF׆��7�tڸ�3���c:�#i>p"�]�:��~p�.�d �"+ %PDF-1.7 rA��E�* ��d�KM (x.a 0000003391 00000 n H�\��j�0��~ A simple approximation of the first derivative is f0(x) ≈ f(x+h)−f(x) h, (5.1) 124 questions for revision practice on methods of differentiation. H��W�۸���V$���(�ۤE���4�"(.EAK�Ĭ$:��>�${�u���k[μy3������c(R'B�.�Go���r7z����Hl�����x%I�l�V�q��l�����Z���b�Y_>��X����~�4Y�������E8�N��>}�o��2^�0��9��G����~����w��z��^�9��_賳�c~�h~�u2�?�8�B���7'K\��C/����y���BX5ڏ��G�f��E�Ç��LJ_a6N�Q,�/b��. That is, if c is a real number, then d dx!c"=0. 0000001156 00000 n startxref {�a�}j���)�_�PUఓAg/f@���Nt��0��9"r������E6ԣ*� Exercise F (trigonometric functions) and Exercise G (implicit functions) complete this package (a pdf on 15 A4 pages). 0000009055 00000 n 0000008266 00000 n 0000002791 00000 n %PDF-1.6 %���� 1 Burns, R (1971), Methods for individualizing instruction. ���8�H The Curriculum advocates the use of a broad range of active learning methodologies such as use of the environment, talk and discussion, collaborative work and use of ICT. 1. �"��{V��M��}zCR�J�b��!����m�tr�r�.ф='�6���`��5�+e��[������?��>ì�ᇁ���h&�4_u�� �6����q��g�d��=Uf����I�d 2 0 obj xref @~ (* {d+��}�G�͋љ���ς�}W�L��$�cGD2�Q���Z4 E@�@����� �A(�q`1���D ������`'�u�4�6pt�c�48.��`�R0��)� Many problems in applied mathematics involve the integration of functions given by complicated formulae, and practi-tioners consult a Table of Integrals in order to complete the integration. 4. 0000046496 00000 n 4 . �㣄������?����G)���� Mathematics / Advanced pure / Differentiation, Standard Form - Notes, Examples and Exercises, A level maths references for university UCAS (updated by strong, middle, weak students), Making links between quadratic formula and graph. Wa�"����Q Conditions. �-{�����Z��aޙ� 2. 0000025638 00000 n Basic Short Cuts for Differentiation ����=�4���@s�9�`4���@g�3�`�x�[)�R�*z���^e��W٫�U�*z���^�V����A��b+�G�Qn��Ji��ƭ[)�Rle�ʰ��o����o����o����o����o��� We will investigate f as x → −1 from below (x < −1) and above (x > −1). 5 acceptent que les garçons interrompent les filles, et s’adressent à eux avec des phrases courtes et directives. 0000028357 00000 n 0000042186 00000 n .��E�#��2�',�$�ň|>>2�&\�p�5�"�B�-5�l���^�5H������$�[���s�� �y�(�a��6�+̬kg��.��-��+�=��利S'a����T.���BSe{j�� �����v���]=���vfpA?��y�e��DS��n��"�T�R����ùiv��Z[���>M�r�r�f�41+�b^���,��8����Tb4�&PE�b���fm��)�:C�eu��LD'1��t�N'�x���4}���B�VM��m���ʠ�;E� Xɑ�߾+0���-I(l!x眗�}�[D�RD@h�k I0�����yR�M�&4�UB͇��� �� |l ye;!�dO��Y$� �1aJ`�WZR)��L��Q,�1X�>��"��_���0L4�B�G�i�^O�ޤ��U��t� �D�S��[��� l�6ٕLI�np�pM�>���������tJ� k.�&$8��OHV%q�/��QU۷�E�N�ܶ�Ed%S�Z}xӲ�M˹��C�,w�I�Ttd�HS��5a�rr�U�!����|u��~]�Rwp��������`xR�R�8D�iz�Ng��ȐӚc���i�L�t��^����d1,T�kB��j H� �6I�>˕T�k���/����>R 0�ʥ�@0�v2}s��uyX�t Z �uM�mLё�%��h`y E,y}PTW��r��q����/dz1���X�h6��mtIb���"r~������ކ2g_��Mh�s�֗��J)a��Yt�/fGXC��b��s�aT( able to come up with methods for approximating the derivatives at these points, and again, this will typically be done using only values that are defined on a lattice. 2 • We have seen two applications: – signal smoothing – root finding • Today we look – differentation – integration • These will form the basis for solving ODEs 0000040762 00000 n Educational Technology, 11, p55-56 2 Zakhartchouck, J.M (2004), Au risque de la pédagogie différenciée, INRP. 0000042539 00000 n Illustration 6 : … 0000001614 00000 n 0000019224 00000 n 0000011234 00000 n 0000025570 00000 n

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