{"id":4080,"date":"2023-10-06T09:27:52","date_gmt":"2023-10-06T09:27:52","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=4080"},"modified":"2023-10-06T09:30:04","modified_gmt":"2023-10-06T09:30:04","slug":"week-4-ss3-second-term-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-4-ss3-second-term-mathematics-notes\/","title":{"rendered":"Week 4 &#8211; SS3 Second Term Mathematics Notes"},"content":{"rendered":"<p>\u00a0<strong>WEEK FOUR<br \/>\n\t\t\t<\/strong><\/p>\n<ul>\n<li>Differentiation of algebraic functions: meaning of differentiation\n<\/li>\n<li>Differentiation from first principle\n<\/li>\n<li>\n<div>Standard derivatives of some basic functions.\n<\/div>\n<p>\u00a0<\/li>\n<\/ul>\n<p>Consider the curve whose equation is given by   y = f(x)  Recall that m = y<sub>2<\/sub> \u2013 y<sub>1<\/sub>= f(x+x)-f(x)<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se1.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se2.png\" alt=\"\"\/>x<sub>2<\/sub>&#8211; x<sub>1<\/sub>x<br \/>\nAs point B moves close to A, dx becomes smaller and tends to zero.<br \/>\nThe limiting value is written on  Lim   f(x +x) \u2013 f(x) and is denoted by as x \u2013&gt; 0<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se3.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0dx<br \/>\nf<sup>l<\/sup>(x) is called the <strong>derivative of f(x)<\/strong> and the <strong>gradient function of the curve<br \/>\n<\/strong><br \/>\n\u00a0The process of finding the derivative of f(x) is called differentiation. The rotations which are commonly used for the derivative of a function are f<sup>1<\/sup>(x) read as f \u2013 prime of x,  df\/dx read as  dee x of f<br \/>\ndf\/dx    read  dee &#8211; f  dee- x,    dy\/dx read  dee &#8211; y  dee- x<\/p>\n<p>\u00a0<strong>If   y = f(x) , this dy\/dx = f<sup>1<\/sup>(x) (it is called the differential coefficient of y with respect to x.<br \/>\n<\/strong><br \/>\n\u00a0<strong>Differentiation from first principle:<\/strong> The process of finding the derivative of a function from the consideration of the limiting value is called differentiation from first principle.<\/p>\n<p>\u00a0<strong>Example 1<br \/>\n<\/strong>Find from first principle, the derivative of   y = x<sup>2<\/sup><br \/>\n\t\tSolution<br \/>\n\u00a0\u00a0\u00a0\u00a0y = x<sup>2<\/sup><br \/>\n\t\t      y + y = (x + x)<sup>2<br \/>\n<\/sup>y + y = x<sup>2<\/sup> + 2xx + (x)<sup>2<\/sup><br \/>\n\t\ty =  x<sup>2<\/sup> + 2xx+ (x)<sup>2 <\/sup> &#8211;  y\u00a0\u00a0\u00a0\u00a0<br \/>\ny = x<sup>2<\/sup>  +  2xx   +  (x)<sup>2<\/sup>  &#8211;  x<sup>2<\/sup><br \/>\n\t\ty  =  2xx  +  (x)<sup>2<\/sup><br \/>\n\t\ty  = (2x   +  x)x<br \/>\ny =  2x   +   x<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se4.png\" alt=\"\"\/>x<br \/>\nLim  x  =  0<br \/>\ndy =  2x<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se5.png\" alt=\"\"\/>dx<br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<br \/>\n<strong>Example 2<\/strong>:<br \/>\nFind from first principle, the derivative of 1\/x<br \/>\nSolution<br \/>\nLet y    =    1<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se6.png\" alt=\"\"\/>x<br \/>\ny + y =      1<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se7.png\" alt=\"\"\/>               x + x<\/p>\n<p>\u00a0y  =       1           &#8211;  y<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se8.png\" alt=\"\"\/>                x + x<br \/>\ny =     1       &#8211;  1<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se9.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se10.png\" alt=\"\"\/>            x + x       x<br \/>\ny  =  x \u2013 (x +  x)<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se11.png\" alt=\"\"\/>               (x  +x)x<br \/>\ny =  x  &#8211;  x  &#8211;  x<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se12.png\" alt=\"\"\/>x<sup>2<\/sup>  +  xx<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se13.png\" alt=\"\"\/>dy  =    -x<br \/>\nx<sup>2<\/sup>+ x<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se14.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se15.png\" alt=\"\"\/>y  =   -1<br \/>\nx     x<sup>2<\/sup> + x<br \/>\nLim  x = 0<br \/>\ndy =    -1<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se16.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se17.png\" alt=\"\"\/>dx         x<sup>2<\/sup><\/p>\n<p>\u00a0<strong>Evaluation: <\/strong>Find from first principle, the derivatives of y with respect to x:<strong><br \/>\n\t\t\t<\/strong><\/p>\n<ol>\n<li>Y = 3x<sup>3<\/sup>                      2. Y = 7x<sup>2<\/sup>     3. Y = 3x<sup>2<\/sup> \u2013 5x\n<\/li>\n<\/ol>\n<p>\u00a0\u00a0\u00a0\u00a0<br \/>\n<strong>Rules of Differentiation:     <\/strong>Let\u00a0\u00a0\u00a0\u00a0y = x<sup>n<\/sup><strong><br \/>\n\t\t\t<\/strong>y + dy = (x + dx)<sup>n<\/sup><br \/>\n\t\t\u00a0\u00a0\u00a0\u00a0= x<sup>n<\/sup> + nx<sup>n-1<\/sup>dx + n(n -1) x<sup>n-2<\/sup>(dx)<sup>2<\/sup> + \u2026 (dx)<sup>n<\/sup><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se18.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0    2!<\/p>\n<p>\u00a0                            = x<sup>n<\/sup> + n x<sup>n-1<\/sup>dx + n(n-1) x<sup>n-2  <\/sup>(dx)<sup>2<\/sup>+ &#8212; + (dx)<sup>n<\/sup> &#8211; x<sup>n<\/sup><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se19.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0        2!<br \/>\n                            = nx<sup>n-1<\/sup>dx + n (n \u2013 1) x<sup>n\u20131 <\/sup>(dx)<sup>2<\/sup><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se20.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02!<br \/>\ndy\/dx = n <sup>xn-1<\/sup> + n (n \u20131) x<sup>n-1 <\/sup>dx<br \/>\nLim dy\/dx = nx<sup>n-1<\/sup><br \/>\n\t\tdx = 0<br \/>\n\u00a0\u00a0\u00a0\u00a0Hence;   <strong>dy\/dx  =  nx<sup>n-1      <\/sup>if y = x<sup>n<\/sup><\/strong><\/p>\n<p>\u00a0<strong>Example 3<\/strong>:\u00a0\u00a0\u00a0\u00a0<br \/>\nFind the derivative of the following with respect to x:   (a) x<sup>7<\/sup> (b) x<sup>\u00bd<\/sup> (c) 5x<sup>2<\/sup> \u2013 3x (d) &#8211; 3x<sup>2<\/sup> (e) y = 2x<sup>3<\/sup> \u2013 3x + 8<br \/>\nSolution<br \/>\na.\u00a0\u00a0\u00a0\u00a0Let  y = x<sup>7<\/sup><br \/>\n\t\t\u00a0\u00a0\u00a0\u00a0dy\/dx = 7 x<sup>7-1 <\/sup>= 7x<sup>6<br \/>\n<\/sup><br \/>\n\u00a0b.\u00a0\u00a0\u00a0\u00a0Let  y = x <sup>\u00bd<\/sup><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week4SS3Se21.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0dy\/dx = \u00bd x<sup>\u00bd -1 <\/sup>= \u00bd x<sup>\u2013 \u00bd<\/sup>  =   1<br \/>\n                                                                  2\u221ax<\/p>\n<p>\u00a0c.            Let y = 5x<sup>2<\/sup> \u2013 3x <sup><br \/>\n\t\t\t<\/sup>dy\/dx = 10x \u2013 3<br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<br \/>\nd.\u00a0\u00a0\u00a0\u00a0Let y = &#8211; 3x<sup>2<\/sup><br \/>\n\t\t\u00a0\u00a0\u00a0\u00a0dy\/dx =2\u00d7 &#8211; 3x<sup>2-1<\/sup> = &#8211; 6x<br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<br \/>\n e.          Let y = 2x<sup>3<\/sup> \u2013 3x + 8<br \/>\n\u00a0\u00a0\u00a0\u00a0dy\/dx= 3 x 2x<sup>3-1 <\/sup>\u2013 3 + 0<br \/>\n= 6x<sup>2<\/sup> \u2013 3 <\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong><\/p>\n<ol>\n<li>If  y=5x<sup>4<\/sup> ,find  dy\/dx        2.Given that y= 4x<sup>-1<\/sup> find y<sup>1<\/sup><strong><br \/>\n\t\t\t\t<\/strong><\/li>\n<\/ol>\n<p>\u00a0<strong>General Evaluation<br \/>\n<\/strong> 1. Find, from first principles, the derivative of  4x<sup>2<\/sup> \u2013 2  with  respect  to  x.<br \/>\n 2. Find the derivative of the following       a.3x<sup>3 <\/sup>\u2013 7x<sup>2<\/sup> \u2013 9x + 4   b. 2x<sup>3<\/sup>    c. 3\/x<br \/>\n\t\t\t<strong>3. <\/strong>Using idea of difference of two square; simplify 243x<sup>2<\/sup> &#8211; 48y<sup>2<strong><br \/>\n\t\t\t\t<\/strong><\/sup>4. Expand (2x -5)( 3x-4)<br \/>\n5. If the gradient of y=2x<sup>2<\/sup>-5 is -12 find the value of y.<\/p>\n<p>\u00a0<strong>Reading Assignment: NGM for SS 3 Chapter 10 <\/strong>page 82 -88, <strong><br \/>\n\t\t\t<\/strong><strong>Weekend Assignment<br \/>\n<\/strong><strong>Objective<br \/>\n<\/strong>1.\u00a0\u00a0\u00a0\u00a0Find the derivative of 5x<sup>3<\/sup>(a) 10x<sup>2<\/sup>     (b) 15x<sup>2<\/sup>      (c) 10x     (d) 15x<sup>3<\/sup><br \/>\n\t\t2.\u00a0\u00a0\u00a0\u00a0Find dy\/dx, if y = 1\/x<sup>3<\/sup>(a) \u20133\/x<sup>4<\/sup>\u00a0\u00a0\u00a0\u00a0(b) 3\/x<sup>4 <\/sup>    (c) 4\/x<sup>3<\/sup>\u00a0\u00a0\u00a0\u00a0(e) \u20134\/x<sup>3<\/sup><br \/>\n\t\t3.\u00a0\u00a0\u00a0\u00a0Find f<sup>1<\/sup>(x), if f(x) = x<sup>3<\/sup> (a) 3x \u00a0\u00a0\u00a0\u00a0(b) 3x<sup>2<\/sup>\u00a0\u00a0\u00a0\u00a0(c) \u00bd x<sup>3<\/sup>\u00a0\u00a0\u00a0\u00a0(d) 2x<sup>3<\/sup><br \/>\n\t\t4.\u00a0\u00a0\u00a0\u00a0Find the derivative of   1\/x(a) 1\/x<sup>2<\/sup>    (b) \u20131\/x<sup>2<\/sup>    (c) \u2013 x \u00a0\u00a0\u00a0\u00a0(d) \u2013x<sup>2<\/sup><br \/>\n\t\t5.\u00a0\u00a0\u00a0\u00a0If      y = &#8211; 2\/3 x<sup>3<\/sup>. Find dy\/dx (a) 4\/3 x<sup>2      <\/sup> (b) 2x<sup>2<\/sup>    (c) \u2013 2x<sup>2        <\/sup>(d) \u20132x<br \/>\n<strong>Theory<br \/>\n\t\t\t<\/strong>1.\u00a0\u00a0\u00a0\u00a0Find from first principle, the derivative of   y = x + 1\/x<br \/>\n2.\u00a0\u00a0\u00a0\u00a0Find the derivative of 2x<sup>2<\/sup> \u2013 2\/x<sup>3<\/sup><\/p>\n<p>\t\t\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0WEEK FOUR Differentiation of algebraic functions: meaning of differentiation Differentiation from first principle Standard derivatives&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,316],"tags":[],"class_list":["post-4080","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss3-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4080","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=4080"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4080\/revisions"}],"predecessor-version":[{"id":4081,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4080\/revisions\/4081"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=4080"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=4080"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=4080"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}