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