If x= {\a} \cos \theta ; {\y}= {\b} \sin \theta, then find { {\d}^2 {\y}}/{ {\d} x^2}. OR Find the differential of \sin ^2 x w.r.t. {\e}^{\cos x}.

CBSE Class 12 Math 2020 Delhi Set 1 Solved Paper - Question 23

Question : 23 of 36

Marks: +1, -0

x= {\a} \cos θ ; {\y}= {\b} sin \;θ

, then find

\;{ {\d}^2 {\y}}/{ {\d} x^2}

.
OR
Find the differential of

sin \;^2 x

w.r.t.

{\e}^{\cos x}

Solution:

OR
Let

{\P}= sin \; {\x}

Q=e^{\cos x}

On differentiating eq.(1) w.r.t. we get

\;{d p}/{d x}=\;{d}/{d x} ( sin \;^2 x) =2 sin \;x \cos x

On differentiating eq.(2) w.r.t we get

\;{ {\d} θ}/{ {\d} x}=\;{ {\d}}/{ {\dx}} ( {\e}^{\cos {\x})}=- {\e}^{\cos {\x}} sin \; {\x}.

dividing eq.(3) and (4)

\;{\;{d p}/{d x}}/{{d θ}/{d x}}=\;{2 sin \;x \cos x}/{-e^{\cos x} sin \;x}=\;{-2 \cos x}/{e^{\cos x}}

{ [\;{\;{d p}/{d x}}/{{d θ}/{d x}}=\;{-2 \cos x}/{e^{\cos x}}] }

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