{"id":3163,"date":"2023-10-04T12:14:35","date_gmt":"2023-10-04T12:14:35","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=3163"},"modified":"2023-10-04T12:23:00","modified_gmt":"2023-10-04T12:23:00","slug":"week-3-ss2-second-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-3-ss2-second-term-further-mathematics-notes\/","title":{"rendered":"Week 3 – SS2 Second Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0WEEK THREE
\n\t\t\t<\/strong>TOPIC :DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS, LOGARITHMIC FUNCTIOS AND EXPONENTIAL FUNCTIONS
\n<\/strong>DERIVATIVE OF TRIGONOMETRIC FUNCTIONS
\nThe derivative of y = sin x dy\/dx = cos x
\nThe derivative of y = cos x dy\/dx = – sin x
\nThe derivative of y= tan x dy\/dx = sec2<\/sup> x\\
\nThe derivative of y = sec x dy\/dx = secxtanx
\nThe derivative of y = cosec x dy\/dx = cosec x cot x
\nThe derivative of y = cot x dy\/dx = – cosec2<\/sup> x
\nExamples
\n(1) If y = cos 2x
\ndy\/dx = – sin2x x d\/dx ( 2x)
\ndy\/dx = -2 sin 2x
\n( 2) If y = cos2<\/sup> x
\nLet u = cos x and y = u2<\/sup>
\n\t\tdy\/du= 2u and du\/dx= -sinx
\ndy\/dx = 2u x \u2013 sinx
\ndy\/dx = – 2 cos x sin x
\n( 3) If y = sec 6x
\nLet u = 6x and y= sec x
\ndu\/dx = 6 and dy\/du = sec u tan u
\ndy\/dx = 6 sec u tan u
\ndy\/dx = 6 sec 6x tan 6x<\/p>\n

\u00a0EVALUATION
\n<\/strong>Differentiate the followings : (i) y = tan 8x (ii) y= cot 5x (iii) y = sin4<\/sup> x<\/p>\n

\u00a0THE DERIVATIVE OF LOGARITHMIC FUNCTIONS
\n<\/strong>If y = loge<\/sub> x dy\/dx = 1\/x(note that loge<\/sub>x=lnx)<\/strong>
\n\t\tExamples
\n<\/strong>(1) If y =loge<\/sub> ( 3x + 2 )
\ndy\/dx =dy\/du x du\/dx
\nLet u = 3x +2 and y = loge<\/sub> u
\ndu\/dx = 3 and dy\/du = 1\/u
\ndy\/du = 1\/u x 3 = 3\/3x + 2
\n(2) If y = loge<\/sub>( 4x \u2013 1) 2<\/sup>
\n\t\tLet u = ( 4x \u2013 1 )2<\/sup> and y = loge<\/sub> u
\ndu\/dx = du\/dv x dv\/dx where v = 4x \u2013 1
\ndu\/dx =2v x 4 = 8v
\ndy\/du = 1\/u
\ndy\/dx =dy\/dx x du\/dx = 1\/u x 8v
\ndy\/dx = 8 (4x \u2013 1)\/(4x \u2013 1)2<\/sup>
\n\t\tdy\/dx = 8 \/4x-1<\/p>\n

\u00a0EVALUATION
\n<\/strong>Differentiate the followings: ( i) y = loge<\/sub> 8x (ii) y = ln ( 6x + 9 )3<\/sup> (iii) y = ln (3x2<\/sup> \u2013 5x +6)<\/p>\n

\u00a0THE DERIVATIVE OF EXPONENTIAL FUNCTIONS
\n<\/strong>If y = ex<\/sup>dy\/dx = ex<\/sup>
\n\t\tExamples
\n<\/strong>(1) If y = e2x<\/sup>
\n\t\tdy\/dx = dy\/du x du\/dx
\nu = 2x and y = eu<\/sup>
\n\t\tdu\/dx = 2 and dy\/du = eu<\/sup>
\n\t\tdy\/dx = eu<\/sup> x 2
\ndy\/dx = e2x<\/sup> x 2
\ndy\/dx = 2 e2x<\/sup>
\n\t\t(2) If y = esin4x<\/sup>
\n\t\tdy\/dx = dy\/du x du\/dx
\nLet u = sin 4x and y = eu
\n<\/sup>du\/dx = 4 cos 4x and dy\/du = eu<\/sup>
\n\t\tdy\/dx =eu<\/sup> x 4 cos 4x
\ndy\/dx = esin x<\/sup>x 4 cos 4x
\ndy\/dx = 4 esin4x<\/sup>cos 4x<\/p>\n

\u00a0EVALUATION
\n<\/strong>Differentiate the followings : (i) y = etan 7x<\/sup> (ii) y = e6x<\/sup> (iii) y = e-5sin3x<\/sup><\/p>\n

\u00a0GENERAL EVALUATION<\/strong>
\n\t\t(1) Find the derivative of each of the following functions : (i) sin3<\/sup> x (ii) cosec x2<\/sup>
\n\t\t(2) Find the derivative of each of the following functions ; (i) log ( x2 <\/sup> -5x + 6 )
\n (3) Differentiate each of the followings : (i) ecosec x <\/sup> (ii) ex<\/sup> – e-x<\/sup>
\n\t\t(4) Differentiate log ( cos x + sin x )
\nReading Assignment : New Further M aths Project 2 page 13o \u2013 137<\/p>\n

\u00a0WEEKEND ASSIGNMENT<\/strong>
\n\t\t1) If y = loge<\/sub> ( 1\/x) find dy\/dx a) 1\/x b) -1\/x c) 1\/x2<\/sup> d) -1\/x2<\/sup>
\n\t\t2) If y = 3 e5x<\/sup> find dy\/dx a) 3e5x<\/sup> b) 15e3x<\/sup> c) 15e5x<\/sup> d) 5e5x<\/sup>
\n\t\t3) If y = sin 4x find dy\/dx a) 4 cos 4x b) -4cos4x c) 4sin4x d) 4tan 4x
\n4) If y = cot 7x finddy\/dx a) 7sec2<\/sup> x b) -7cosec2<\/sup> x c) -7cosec2<\/sup> 7x d) 7 tan 7x
\n5) Differentiate sin x \u2013 cos x a) sinx + cosx b) cosx \u2013 sinx c) sinx- cosx d) -sinx-cosx<\/p>\n

\u00a0THEORY
\n<\/strong>1) Differentiate the followings ; (i) cos3<\/sup> x (ii) sin 4x (iii) ecos 5x<\/sup> (iv) cos4x3<\/sup>
\n\t\t2) Differentiate the followings : (i) ln sin x (ii) log ( x2<\/sup> \u2013 2) (iii) log ( 1 + x )4<\/sup><\/p>\n

\u00a0
\n\t\t\t<\/strong>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

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