{"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 Question 1. Question 2. <\/p>\n Question 3. Question 4. Question 5. Question 6. Question 7. <\/p>\n Question 8. Question 9. Question 10. Question 11. Question 12. Question 13. <\/p>\n Question 14. Question 15. Question 16. Question 17. Question 18. Question 19. <\/p>\n Question 20. Question 21. Question 22. Question 23. <\/p>\n Question 24. 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}]}}TS Inter 1st Year Maths 1B Differentiation Important Questions Short Answer Type<\/h2>\n
\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<\/p>\n
\nFind the derivative of \\(\\sqrt{x+1}\\) from the first principle. [Mar. ’12, ’05]
\nSolution:
\n
\n<\/p>\n
\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
\n<\/p>\n
\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<\/p>\n
\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
\n<\/p>\n
\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
\n<\/p>\n
\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
\n<\/p>\n
\nFind the derivative of cos2<\/sup>x from the first principle. [Mar. ’19 (TS); May ’08, ’04]
\nSolution:
\n<\/p>\n
\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
\n\u2234 f is differentiable at x and f'(x) = \\(\\frac{1}{x}\\)<\/p>\n
\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
\n
\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
\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
\n<\/p>\n
\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
\n<\/p>\n
\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<\/p>\n
\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
\n<\/p>\n
\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
\n<\/p>\n
\nFind the derivative of tan-1<\/sup>(sec x + tan x). [May ’97]
\nSolution:
\n
\n<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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
\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
\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
\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<\/p>\n
\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
\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
\n<\/p>\n
\nFind the derivative of cot x from the first principle. [Mar. ’19, ’17 (AP)]
\nSolution:
\n<\/p>\n","protected":false},"excerpt":{"rendered":"