Differentiate (1+x^2 ) with respect to a^x.

Question

Differentiate $\log \left(1+x^2\right)$ with respect to $a^x$.

✓

Answer

Let $u =\log \left(1+ x ^2\right)$ and $v = a ^{ x }$
Then we want to find $\frac{d u}{d v}$
Differentiating $u$ and $v$ w.r.t. $x$, we get
$
\begin{aligned}
\frac{d u}{d x} & =\frac{d}{d x}\left[\log \left(1+x^2\right)\right]=\frac{1}{1+x^2} \cdot \frac{d}{d x}\left(1+x^2\right) \\
& =\frac{1}{1+x^2} \times(0+2 x)=\frac{2 x}{1+x^2}
\end{aligned}
$
and $\frac{d v}{d x}=\frac{d}{d x}\left(a^x\right)=a^x \cdot \log a$
$
\therefore \frac{d u}{d v}=\frac{(d u / d x)}{(d v / d x)}=\frac{\left(\frac{2 x}{1+x^2}\right)}{a^x \cdot \log a}=\frac{2 x}{\left(1+x^2\right) \cdot a^x \log a} \text {. }
$

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