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Differentiate the given function w.r.t x:${\left( {\sin x - \cos x} \right)^{\left( {\sin x - \cos x} \right)}},\dfrac{\pi }{4} < x < \dfrac{{3\pi }}{4}$

Last updated date: 30th May 2024
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Hint: As the function is in the form of variable to the power of variable we apply log on both sides of the equation and then differentiate.

Let $y = {\left( {\sin x - \cos x} \right)^{\left( {\sin x - \cos x} \right)}}$
Take log both the side
$\Rightarrow \log y = \log {\left( {\sin x - \cos x} \right)^{\left( {\sin x - \cos x} \right)}}$
We know that $\log {a^b} = b\log a$
$\Rightarrow \log y = \left( {\sin x - \cos x} \right)\log \left( {\sin x - \cos x} \right)$
Now differentiate both the side w.r.t x
Here we use chain rule of differentiation
Differentiation of sinx wrt x is cosx
Differentiation of cosx wrt x is -sinx
Differentiation of logx wrt x is $\dfrac{1}{x}$

$\Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \left( {\sin x - \cos x} \right)\dfrac{{d\log \left( {\sin x - \cos x} \right)}}{{dx}} + \log \left( {\sin x - \cos x} \right)\dfrac{{d\left( {sinx - \cos x} \right)}}{{dx}}$
$\Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \left( {\sin x - \cos x} \right)\dfrac{1}{{\left( {\sin x - \cos x} \right)}}\dfrac{{d\left( {\sin x - \cos x} \right)}}{{dx}} + \log \left( {\sin x - \cos x} \right)\left( {\cos x + \sin x} \right)$
$\Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \left( {\sin x - \cos x} \right)\dfrac{1}{{\left( {\sin x - \cos x} \right)}}\left( {\cos x + \sin x} \right) + \log \left( {\sin x - \cos x} \right)\left( {\cos x + \sin x} \right)$
$\Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \left( {\cos x + \sin x} \right) + \log \left( {\sin x - \cos x} \right)\left( {\cos x + \sin x} \right)$
$\Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \left( {1 + log\left( {\sin x - \cos x} \right)} \right)\left( {\cos x + \sin x} \right)$
$\Rightarrow \dfrac{{dy}}{{dx}} = y\left( {1 + log\left( {\sin x - \cos x} \right)} \right)\left( {\cos x + \sin x} \right)$
$\Rightarrow \dfrac{{dy}}{{dx}} = {\left( {\sin x - \cos x} \right)^{\left( {\sin x - \cos x} \right)}}\left( {1 + log\left( {\sin x - \cos x} \right)} \right)\left( {\cos x + \sin x} \right)$
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