Maharashtra State BoardHSC Commerce 12th Board Exam
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Find d2ydx2, if y = x2⋅ex - Mathematics and Statistics

Sum

Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`

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Solution

y = `"x"^2 * "e"^"x"`

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

`"dy"/"dx" = "x"^2 * "d"/"dx" ("e"^"x") * "d"/"dx" ("x"^2)`

`= "x"^2 * "e"^"x" + "e"^"x" ("2x")`

`"dy"/"dx" = ("x"^2 + 2"x") * "e"^"x"`

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

`("d"^2"y")/"dx"^2 = ("x"^2 + 2"x") * "d"/"dx" ("e"^"x") + "e"^"x" * "d"/"dx" ("x"^2 + 2"x")`

`= ("x"^2 + 2"x") * "e"^"x" + "e"^"x"`(2x + 2)

`= "e"^"x"("x"^2 + 2"x" + 2"x" + 2)`

∴ `("d"^2"y")/"dx"^2 = "e"^"x"("x"^2 + 4"x" + 2)`

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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 3 Differentiation
Miscellaneous Exercise 3 | Q 4.21 | Page 101
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