Differentiate the following w. r. t. x.y=a^ a^ _0( x) - Vidyadip

Question

Differentiate the following w. r. t. x.$y=a^{a^{\log _0(\cos x)}}$

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Answer

$
\begin{aligned}
& y=a^{a^{\log _a(\operatorname{soc} x)}} \\
& y=a^{\cot x} \quad\left[\because a^{\log _a f(x)}=f(x)\right]
\end{aligned}
$
Differentiate $w . r . t . x$
$
\begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left(a^{\cot x}\right) \\
& =a^{\cot x} \log a \cdot \frac{d}{d x}(\cot x) \\
& =a^{\cot x} \log a\left(-\operatorname{cosec}^2 x\right) \\
\frac{d y}{d x} & =-\operatorname{cosec}^2 x \cdot a^{\cot x} \log a
\end{aligned}
$

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