Maharashtra State BoardHSC Science (General) 11th
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Solve the following problem. Using the rule for differentiation for quotient of two functions, prove that ddx(sin xcos x)=sec2x - Physics

Sum

Solve the following problem.

Using the rule for differentiation for quotient of two functions, prove that `"d"/"dx" ("sin x"/"cos x") = sec^2"x"`. 

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Solution

Using, `"d"/"dx"[("f"_1("x"))/("f"_2("x"))] = 1/("f"_2("x")) ("df"_1("x"))/"dx" - ("f"_1("x"))/("f"_2^2("x")) ("df"_2("x"))/"dx"`

For f1(x) = sin x and f2(x) = cos x

`"d"/"dx" ("sin x"/"cos x") = 1/"cos x" xx ("d"(sin "x"))/"dx" - "sin x"/cos^2"x" xx ("d"(cos "x"))/"dx"`

`= 1/"cos x" xx "cos x" - "sin x"/(cos^2"x") xx (- sin "x")`

`= 1 + (sin^2"x")/(cos^2"x") = (cos^2"x" + sin^2"x")/(cos^2"x")`

∴ `"d"/"dx"("sin x"/"cos x") = 1/(cos^2"x")  ....[sin^2"x" + cos^2"x" = 1]`

= sec2x      ....`(because 1/"cos x" = sec "x")`

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APPEARS IN

Balbharati Physics 11th Standard Maharashtra State Board
Chapter 2 Mathematical Methods
Exercises | Q 3. (vi) | Page 29
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