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Question
If x = t2, y = t3, then `("d"^2"y")/("dx"^2)` is ______.
Options
`3/2`
`3/(4"t")`
`3/(2"t")`
`3/4`
Solution
If x = t2, y = t3, then `("d"^2"y")/("dx"^2)` is `3/(4"t")`.
Explanation:
Given that x = t2 and y = t3
Differentiating both the parametric functions w.r.t. t
`"dx"/"dt"` = 2t and `"dy"/"dt"` = 3t2
∴ `"dy"/"dx" = ("dy"/"dt")/("dx"/"dt")`
= `(3"t"^2)/(2"t")`
= `3/2 "t"`
⇒ `"dy"/"dx" = 3/2 "t"`
Now differentiating again w.r.t. x
`"d"/"dx"("dy"/"dx") = 3/2 * "dt"/"dx"`
⇒ `("d"^2"y")/("dx"^2) = 3/2 * 1/(2"t")`
= `3/(4"t")`.
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