Find d y d x if, : y=a^ (1+ x) - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=a^{(1+\log x)}
$

✓

Answer

Given : $y=a^{(1+\log x)}$
Let $u=1+\log x$
Then $y=a^u$
$
\begin{aligned}
\therefore \frac{d y}{d u} & =\frac{d}{d u}\left(a^u\right)=a^u \cdot \log a \\
& =a^{(1+\log x)} \cdot \log a
\end{aligned}
$
and $\frac{d u}{d x}=\frac{d}{d x}(1+\log x)$
$
=0+\frac{1}{x}=\frac{1}{x}
$
$
\begin{aligned}
\therefore \frac{d y}{d x} & =\frac{d y}{d u} \cdot \frac{d u}{d x} \\
& =a^{(1+\log x)} \cdot \log a \cdot \frac{1}{x} .
\end{aligned}
$

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