Question

logarithmic differenciation y=sinx rais to x​

Answer 1

Hey !

Given,

$\mathbf{\underline{y = (\sin\, x)^{x}}}$


Apply natural log. ln on both sides

$ln \: y = ln\: (\sin\, x)^{x} \:$


From logarithmic property, we have

$\huge{\fbox{\mathbf{log\, a^{b} = b\cdot log\, a}}} \:$


$\implies \: ln \: y =x \, ln\: (\sin\, x) \:$


Differentiate on both sides wrt x

Here y is a f(x) , so y is another variable.


Apply uv rule for RHS

$\huge{\fbox{\mathbf{d{uv} = vu' + uv'}}} \:$


where u and v are function of same variables and

u' and v' are it's derivatives respectively


$\implies \dfrac{1}{y} \dfrac{dy}{dx} =[\cos \, x\, \dfrac{1}{\sin\, x}] \cdot x + 1 \cdot. ln\, (\sin\, x) \\ \\ \implies \dfrac{1}{y} \dfrac{dy}{dx} = x \cot\, x + ln\, (sin\, x) \\ \\ \implies \dfrac{dy}{dx} = x \cot\, x + ln\, (sin\, x) \times y$

Substitute y in RHS


$\underline{\underline{\implies \mathsf{\dfrac{dy}{dx} = \left(x \cot\, x + ln\, (sin\, x)\right) \cdot [(\sin\, x)^{x}]}}}$



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