Calculus: Differentiation and Applications of Derivatives

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Calculus: Differentiation and Applications of Derivatives1.Differentiate the following with respect to ‘x’a)y = x^4–3x^2 + 2b)y = 3x–5x^3c)y = 6 / squareroot x–4 squareroot xd)y = (x+5) (2x-1)(Criteria to meet: Differentiate from 1stprinciple)Solution:a.

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()()()()()()()424242424232(1)Taking changeinandin,32323232by (1)yxxxxyyyyxxxxyxxxxyyxxxxxx=−++=+−++=+−++−=+−++−−+()() ()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()4224222242222222242243324232324234432Since22222222224464xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx+=+++=+++++=+++++++++=+++++++++=++++()()()()()()()()()()()()()()23424322422342432242Therefore46432232464336232yxxxxxxxxxxxxxxyxxxxxxxxxxxxxx=++++−+++−−+=++++−−−+−+−()()()()()()23423246436yxxxxxxxxxx=+++−−()()()()()()023423200Dividing both sides byand taikinglim46436limlimxxxxxxxxxxxxxxyxx→→→+++−−=()()()()()()()()()()()()()()()2332002332002332346436limlimlimlim464364604003 0646xxxxxxxxxxxxxyxxyxxxxxxxxxdyxxxxdxdyxxdx→→→→+++−−==+++−−=+++−−=−

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b.()()()()()()()3333335(1)Taking changeinandin,35353535by (1)yxxxxyyyyxxxxyxxxxyyxxxxxx=−+=+−+=+−+−=+−+−−()()()()()()()()()()()32323323233223533353355151535351515yxxxxxxxxxxyxxxxxxxxxxyxxxxxx=+−+++−+=+−−−−−+=−−−()()()()()()0322002200Dividing both sides byand taikinglim351515limlim351515limlimxxxxxxxxxxxxyxxxxxxxyxx→→→→→−−−=−−−=()()()22235 015015315dyxxdxdyxdx=−−−=−c.

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