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Q} Differentiate : x^sinx with respect to (sinx)^x

Q} Differentiate :
 
x^sinx  with respect to   (sinx)^x

Grade:12

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
9 years ago
Dear student,
We have:
u=x^{\sin x} \Rightarrow \log u=\sin x \log x\Rightarrow \frac{1}{u}\frac{du}{dx}=\cos x\log x+\frac{\sin x}{x}\Rightarrow \frac{du}{dx}=x^{\sin x}\log x^{\cos x}+x^{\sin x}\frac{\sin x}{x}SImilarly,
v=\left (\sin x \right )^{x} \Rightarrow \log v=x \log\sin x\Rightarrow \frac{1}{v}\frac{dv}{dx}=\log\sin x+\frac{x}{\tan x}\Rightarrow \frac{dv}{dx}=\left (\sin x \right )^x\log\sin x+x\left (\sin x \right )^x \cot xHence, taking the ratio, would give the required result.
Regards

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