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Continuity and Differentiability

Short Answer Type

271. Differentiate the following w.r.t. x :
x^x + x^a + a^x + a^a for some fixed a > 0 and x > 0

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272.

Find space dy over dx comma space if space straight y to the power of straight x plus straight x to the power of straight y plus straight x to the power of straight x equals straight a to the power of straight b.

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273.

If space straight y equals 1 plus fraction numerator straight c subscript 1 over denominator straight x minus straight c subscript 1 end fraction plus fraction numerator straight c subscript 2 straight x over denominator left parenthesis straight x minus straight c subscript 1 right parenthesis left parenthesis straight x minus straight c subscript 2 right parenthesis end fraction comma space prove space that space dy over dx equals straight y over straight x open parentheses fraction numerator straight c subscript 1 over denominator straight c subscript 1 minus straight x end fraction plus fraction numerator straight c subscript 2 over denominator straight c subscript 2 minus straight x end fraction close parentheses

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274.

If space straight y equals fraction numerator ax over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator bx over denominator left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction plus fraction numerator straight c over denominator straight x minus straight c end fraction plus 1. prove space that space fraction numerator straight y apostrophe over denominator straight y end fraction equals 1 over straight x open parentheses fraction numerator straight a over denominator straight a minus straight x end fraction plus fraction numerator straight b over denominator straight b minus straight x end fraction plus fraction numerator straight c over denominator straight c minus straight x end fraction close parentheses

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275. If u, v, w are differentiable function ofx, then show that

straight d over dx left parenthesis straight u. straight v. straight w right parenthesis equals du over dx. vw plus straight u. dv over dx. straight w plus straight u. straight v dw over dx

in two ways - first by repeated application of product rule, second by logarithmic differentiation.

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276.
Differentiate (x²-5x+8)(x³+7x+ 9) in three ways mentioned below :
(i) by using product rule.
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Also verify that three answers so obtained are the same.

Let space space space space space space space space space straight y equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis left parenthesis straight i right parenthesis therefore space dy over dx equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis. straight d over dx left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis plus left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis. straight d over dx left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis space space space space space space equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis 3 straight x squared plus 7 right parenthesis plus left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis left parenthesis 2 straight x minus 5 right parenthesis space space space space space space equals 3 straight x to the power of 4 plus 7 straight x squared minus 15 straight x cubed minus 35 straight x plus 24 straight x squared plus 56 plus 2 straight x to the power of 4 minus 5 straight x cubed plus 14 straight x squared minus 35 straight x plus 18 straight x minus 45 space space space space space space equals 5 straight x to the power of 4 minus 20 straight x cubed plus 45 straight x squared minus 5 straight x minus 52 straight x plus 11 left parenthesis ii right parenthesis space space space space space space space space space space space straight y equals straight x to the power of 5 minus 5 straight x to the power of 4 plus 15 straight x cubed minus 26 straight x squared plus 11 straight x plus 72 therefore 1 over straight y dy over dx equals 5 straight x to the power of 4 minus 20 straight x 3 plus 45 straight x squared minus 52 straight x plus 11 left parenthesis iii right parenthesis space space space space log space straight y equals log left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis plus log left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis therefore space 1 over straight y dy over dx equals fraction numerator 2 straight x minus 5 over denominator straight x squared minus 5 space straight x plus 8 end fraction plus fraction numerator 3 straight x squared plus 7 over denominator straight x cubed plus 7 straight x plus 9 end fraction therefore space space space space space space space dy over dx equals left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis open parentheses fraction numerator 2 straight x minus 5 over denominator straight x squared minus 5 space straight x plus 8 end fraction plus fraction numerator 3 straight x squared plus 7 over denominator straight x cubed plus 7 straight x plus 9 end fraction close parentheses space space space space space space space space space space space space space space space space space space space equals left parenthesis 2 straight x minus 5 right parenthesis left parenthesis straight x cubed plus 7 straight x plus 9 right parenthesis plus left parenthesis 3 straight x squared plus 7 right parenthesis left parenthesis straight x squared minus 5 space straight x plus 8 right parenthesis space space space space space space space space space space space space space space space space space space space equals 5 straight x to the power of 4 minus 20 straight x cubed plus 45 straight x squared minus 52 straight x plus 11 Answer space in space the space all space the space cases space is space same.

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277. Use Chain Rule to differentiate sin (x²) w.r.t.x.

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278. Use Chain Rule to differentiate w.r.t.x: tan (2 x + 3)

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279. Use Chain Rule to differentiate w.r.t.x: sin (cos x²)

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280. Differentiate w.r.t.x:

sin left parenthesis straight x squared plus 5 right parenthesis

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

276. Differentiate (x2-5x+8)(x3+7x+ 9) in three ways mentioned below :(i) by using product rule.(ii) by expanding the product to obtain a single polynomial.(iii) by logarithmic differentiation.Also verify that three answers so obtained are the same.

276.
Differentiate (x²-5x+8)(x³+7x+ 9) in three ways mentioned below :
(i) by using product rule.
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Also verify that three answers so obtained are the same.