Find d y d x if y =2^ x^x . - Vidyadip

Question

Find $\frac{d y}{d x}$ if $y =2^{x^x}$.

✓

Answer

Given : $y=2^{x^x}$
Let $u=x^x$
Then $y=2^u$
$
\begin{aligned}
\therefore \frac{d y}{d u} & =\frac{d}{d u}\left(2^u\right)=2^u \cdot \log 2 \\
& =2^{x^x} \cdot \log 2
\end{aligned}
$
Now, $u=x^x$
$
\therefore \log u=\log x^x=x \log x
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
\frac{1}{u} \cdot \frac{d u}{d x} & =\frac{d}{d x}(x \log x) \\
& =x \frac{d}{d x}(\log x)+(\log x) \cdot \frac{d}{d x}(x) \\
& =x \times \frac{1}{x}+(\log x) \times 1 \\
\therefore \frac{d u}{d x} & =u(1+\log x)=x^x(1+\log x) \\
\therefore \frac{d y}{d x} & =\frac{d y}{d u} \cdot \frac{d u}{d x} \\
& =2^{x^x} \cdot \log 2 \cdot x^x(1+\log x) \\
& =2^{x^x} \cdot x^x(\log 2)(1+\log x) .
\end{aligned}
$
... [By (1) and (2)]

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