A) \[\log y=\tan x\frac{dy}{dx}\]
B) \[y\log y=\tan x\frac{dy}{dx}\]
C) \[y\log y=\sin x\frac{dy}{dx}\]
D) \[\log y=\cos x\frac{dy}{dx}\]
E) \[y\log y=\cos x\frac{dy}{dx}\]
Correct Answer: B
Solution :
Given curve is \[y={{e}^{a\sin x}}\] ...(i) Taking log on both sides, we get \[log\text{ }y=a\text{ }sin\text{ }x\] ...(ii) Differentiating w.r.t.\[x,\]we get \[\frac{1}{y}\frac{dy}{dx}=a\cos x\] ...(iii) Dividing Eq. (iii) by Eq. (ii), we get \[\frac{\frac{1}{y}\frac{dy}{dx}}{\log y}=\frac{a\cos x}{a\sin x}\] \[\Rightarrow \] \[\frac{dy}{dx}=y\log y\cot x\] \[\Rightarrow \] \[y\log y=\tan x\frac{dy}{dx}\]
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