{"id":4930,"date":"2024-01-03T11:40:14","date_gmt":"2024-01-03T06:10:14","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=4930"},"modified":"2024-01-05T17:47:55","modified_gmt":"2024-01-05T12:17:55","slug":"maths-1b-differentiation-important-questions-short-answer-type","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/maths-1b-differentiation-important-questions-short-answer-type\/","title":{"rendered":"TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type"},"content":{"rendered":"

Students must practice these Maths 1B Important Questions<\/a> TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type to help strengthen their preparations for exams.<\/p>\n

TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type<\/h2>\n

Question 1.
\nFind the derivative of x3<\/sup> from the first principle. [Mar. ’15 (TS), ’98]
\nSolution:
\nLet f(x) = x3<\/sup> then f(x + h) = (x + h)3<\/sup>
\n\"TS<\/p>\n

Question 2.
\nFind the derivative of \\(\\sqrt{x+1}\\) from the first principle. [Mar. ’12, ’05]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 3.
\nFind the derivative of sin 2x from the first principle. [B.P. May ’15 (TS), ’10, ’91; Mar. ’02; Mar. ’18 (AP)]
\nSolution:
\nLet f(x) = sin 2x
\nf(x + h) = sin 2(x + h) = sin 2x + 2h
\n\"TS
\n\"TS<\/p>\n

Question 4.
\nFind the derivative of cos ax from the first principle. [May ’14; Mar. ’13 (old), ’13, ’11, ’09]
\nSolution:
\nLet f(x) = cos ax
\nf(x + h) = cos a(x + h) = cos(ax + ah)
\n\"TS<\/p>\n

Question 5.
\nFind the derivative of tan 2x from the first principle. [Mar. ’14, ’13 (old). ’05; May ’13. ’11; May ’15 (AP)]
\nSolution:
\nLet f(x) = tan 2x
\nf(x + h) = tan 2(x + h) = tan (2x + 2h)
\n\"TS
\n\"TS<\/p>\n

Question 6.
\nFind the derivative of sec 3x from the first principle. [Mar. ’16 (AP), ’12, ’08]
\nSolution:
\nLet f(x) = sec 3x
\nf(x + h) = sec 3(x + h) = sec (3x + 3h)
\n\"TS
\n\"TS<\/p>\n

Question 7.
\nFind the derivative of x sin x from the first principle. [Mar. ’18, ’15 (AP), ’10; May ’09]
\nSolution:
\nLet f(x) = x sin x
\nf(x + h) = (x + h) sin (x + h)
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 8.
\nFind the derivative of cos2<\/sup>x from the first principle. [Mar. ’19 (TS); May ’08, ’04]
\nSolution:
\n\"TS<\/p>\n

Question 9.
\nFind the derivative of log x from the first principle. [Mar. ’03]
\nSolution:
\nGiven, f(x) = log x
\nNow, f(x + h) = log (x + h)
\n\"TS
\n\u2234 f is differentiable at x and f'(x) = \\(\\frac{1}{x}\\)<\/p>\n

Question 10.
\nProve that \\(\\frac{d}{d x} \\mathbf{u v}=\\mathbf{u} \\frac{d v}{d x}+v \\frac{d u}{d x}\\). [May ’97]
\n(Or)
\nIf f, g are two differentiable functions at x then fg is differentiable at x. then show that (fg)’ (x) = f(x) g'(x) + g(x) f'(x).
\nSolution:
\nSince f and g are two differentiable functions at x, f'(x) and g'(x) exist.
\n\"TS
\n\"TS
\n= f(x + 0) . g'(x) + g(x) . f'(x)
\n= f(x) . g'(x) + g(x) . f'(x)
\n\u2234 fg is differentiable at x and (fg)’ (x) = f(x) g'(x) + g(x) f'(x).<\/p>\n

Question 11.
\nProve that \\(\\frac{d}{d x}\\left(\\frac{u}{v}\\right)=\\frac{v \\frac{d u}{d x}-u \\frac{d v}{d x}}{v^2}\\). [May ’04, ’98]
\n(Or)
\nIf f, g are two differentiable functions at x and g(x) \u2260 0 then \\(\\frac{f}{g}\\) is differentiable at x, then show that \\(\\left(\\frac{f}{g}\\right)^{\\prime}(x)=\\frac{g(x) f^{\\prime}(x)-f(x) g^{\\prime}(x)}{[g(x)]^2}\\)
\nSolution:
\nSince f, g are differentiable at x and f'(x), g'(x) exists.
\n\"TS
\n\"TS<\/p>\n

Question 12.
\nFind the derivative of \\(\\cos ^{-1}\\left(\\frac{b+a \\cos x}{a+b \\cos x}\\right)\\), (a > 0, b > 0). [May ’09]
\nSolution:
\nLet y = \\(\\cos ^{-1}\\left(\\frac{b+a \\cos x}{a+b \\cos x}\\right)\\)
\nDifferentiating on both sides with respect to ‘r’ to ‘x’
\n\"TS
\n\"TS<\/p>\n

Question 13.
\nIf xy = ex-y, then show that \\(\\frac{d y}{d x}=\\frac{\\log x}{(1+\\log x)^2}\\). [Mar. ’08, ’07, ’96, ’88; May ’00, ’95]
\nSolution:
\nGiven, xy<\/sup> = ex-y<\/sup>
\nTaking logarithms on both sides,
\nlog(xy<\/sup>) = log(ex-y<\/sup>)
\ny log x = (x – y) log e
\ny log x = x – y
\ny + y log x = x
\ny(1 + log x) = x
\ny = \\(\\frac{x}{1+\\log x}\\)
\nDifferentiating on both sides with respect to ‘x’
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 14.
\nIf sin y = x sin (a + y), then show that \\(\\frac{d y}{d x}=\\frac{\\sin ^2(a+y)}{\\sin a}\\). [Mar. ’95, May ’87]
\nSolution:
\nGiven, sin y = x sin (a + y)
\nx = \\(\\frac{\\sin y}{\\sin (a+y)}\\)
\nDifferentiating on both sides with respect to ‘x’.
\n\"TS
\n\"TS<\/p>\n

Question 15.
\nDifferentiate \\(\\tan ^{-1}\\left(\\frac{2 x}{1-x^2}\\right)\\) with respect to \\(\\sin ^{-1}\\left(\\frac{2 x}{1+x^2}\\right)\\). [Mar. ’13 (Old); May ’04, ’95]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 16.
\nFind the derivative of tan-1<\/sup>(sec x + tan x). [May ’97]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 17.
\nIf x = a (cos t + t sin t), y = a (sin t – t cos t) then find \\(\\frac{d \\mathbf{y}}{d \\mathbf{x}}\\). [Mar. ’16 (TS), May ’08, ’00, ’93]
\nSolution:
\nGiven that x = a(cos t + t sin t)
\nDifferentiating on both sides with respect to ‘t’.
\n\"TS<\/p>\n

Question 18.
\nIf x = a(t – sin t), y = a(1 + cos t) then find \\(\\frac{d^2 \\mathbf{y}}{\\mathbf{d x}^2}\\). [May ’02]
\nSolution:
\nGiven, x = a(t – sin t)
\nDifferentiating on both sides with respect to ‘x’.
\n\\(\\frac{\\mathrm{dx}}{\\mathrm{dt}}\\) = a(1 – cos t)
\ny = a(1 + cos t)
\nDifferentiating on both sides with respect to ‘x’.
\n\\(\\frac{\\mathrm{dy}}{\\mathrm{dt}}\\) = a(0 – sin t) = -a sin t
\n\"TS<\/p>\n

Question 19.
\nFind the second order derivative of \\(\\tan ^{-1}\\left(\\frac{1+x}{1-x}\\right)\\). [May ’12]
\nSolution:
\nGiven, f(x) = \\(\\tan ^{-1}\\left(\\frac{1+x}{1-x}\\right)\\)
\nDifferentiating on both sides with respect to ‘x’.
\n\"TS
\nAgain differentiating on both sides with respect to ‘x’.
\nf”(x) = \\(-\\frac{1}{\\left(1+x^2\\right)^2}(0+2 x)=\\frac{-2 x}{\\left(1+x^2\\right)^2}\\)<\/p>\n

\"TS<\/p>\n

Question 20.
\nIf y = aenx<\/sup> + be-nx<\/sup>, then prove that y” = n2y. [Mar. ’15 (AP); May ’14]
\nSolution:
\nGiven y = aenx<\/sup> + be-nx<\/sup> ………(1)
\nDifferentiating on both sides with respect to ‘x’.
\ny’ = aenx<\/sup> (n) + be-nx<\/sup> (-n)
\ny’ = n aenx<\/sup> – n be-nx<\/sup>
\nagain differentiating on both sides with respect to ‘x’
\ny” = na . enx<\/sup> (n) – n be-nx<\/sup> (-n)
\n= n2<\/sup> aexnx<\/sup> + n2be-nx<\/sup>
\n= n2<\/sup> (aenx<\/sup> + be-nx<\/sup>)
\n= n2<\/sup>y (\u2235 from 1)
\n\u2234 y” = n2<\/sup>y<\/p>\n

Question 21.
\nIf y = axn+1<\/sup> + bx-n<\/sup> then prove that x2<\/sup>y11<\/sup> = n(n + 1)y. [May ’10; Mar. ’06]
\nSolution:
\nGiven, y = axn+1<\/sup> + bx-n<\/sup> …….(1)
\nDifferentiating on both sides with respect to ‘x’.
\ny1<\/sup> = a . (n + 1) xn+1-1<\/sup> + b(-n) x-n-1<\/sup>
\n= a(n + 1)xn<\/sup> – bn . x-n-1<\/sup>
\nAgain differentiating on both sides with respect to ‘x’.
\n\"TS<\/p>\n

Question 22.
\nIf y = a cos x + (b + 2x) sin x, then prove that y11<\/sup> + y = 4 cos x. [May ’07]
\nSolution:
\nGiven, y = a cos x + (b + 2x) sin x ………(1)
\nDifferentiating on both sides with respect to ‘x’.
\ny1<\/sup> = a(-sin x) + (b + 2x) cos x + sin x (0 + 2 . 1)
\ny1<\/sup> = -a sin x + (b + 2x) cos x + 2 sin x
\nAgain differentiating of both sides with respect to ‘x’.
\ny11<\/sup> = -a cos x + (b + 2x) (-sin x) + cos x (0 + 2 . 1) + 2 cos x
\n= -a cos x – (b + 2x) sin x + 2 cos x + 2 cos x
\n= -a cos x – (b + 2x) sin x + 4 cos x
\n= -[a cos x + (b + 2x) sin x] + 4 cos x
\ny11<\/sup>\u00a0= -y + 4 cos x [\u2235 from (1)]
\ny11<\/sup> + y = 4 cos x<\/p>\n

Question 23.
\nIf ax2<\/sup> + 2hxy + by2<\/sup> = 1 then prove that \\(\\frac{d^2 y}{d x^2}=\\frac{h^2-a b}{(h x+b y)^3}\\). [Mar. ’08; May ’97]
\nSolution:
\nGiven, ax2<\/sup> + 2hxy + by2<\/sup> = 1 ……..(1)
\nDifferentiating on both sides with respect to ‘x’.
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 24.
\nFind the derivative of cot x from the first principle. [Mar. ’19, ’17 (AP)]
\nSolution:
\n\"TS<\/p>\n","protected":false},"excerpt":{"rendered":"

Students must practice these Maths 1B Important Questions TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type to help strengthen their preparations for exams. TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type Question 1. Find the derivative of x3 from the first principle. [Mar. ’15 (TS), ’98] Solution: … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4930"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=4930"}],"version-history":[{"count":3,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4930\/revisions"}],"predecessor-version":[{"id":5713,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4930\/revisions\/5713"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=4930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=4930"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=4930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}