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Find d2ydx2, if y = 2at, x = at2

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Question

Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2

Sum
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Solution

x = at

Differentiating both sides w.r.t. t, we get

`"dx"/"dt" = "a" "d"/"dx" ("t"^2) = "a"("2t")`

∴ `"dx"/"dt" = "2at"`      ....(i)

y = 2at

Differentiating both sides w.r.t. t, we get

`"dy"/"dt" = "2a" "d"/"dt" ("t")`

∴ `"dy"/"dt"` = 2a

∴ `"dy"/"dx" = ("dy"/"dt")/("dx"/"dt") = "2a"/"2at" = 1/"t"`

Again, differentiating both sides w.r.t. x, we get

`("d"^2"y")/"dx"^2 = (-1)/"t"^2 * "dt"/"dx" = (-1)/"t"^2 xx 1/"2at"`     ....[From (i)]

`= (- 1)/"2at"^3`

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Chapter 3: Differentiation - MISCELLANEOUS EXERCISE - 3 [Page 101]

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Balbharati Mathematics and Statistics 1 (Commerce) [English] Standard 12 Maharashtra State Board
Chapter 3 Differentiation
MISCELLANEOUS EXERCISE - 3 | Q IV] 20) | Page 101

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