Differentiate e ^ (4 x+5) with resepct to 10^ 4 x .

Question

Differentiate $e ^{(4 x+5)}$ with resepct to $10^{4 x}$.

✓

Answer

Let $u=e^{(4 x+5)}$ and $v=10^{4 x}$
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[e^{(4 x+5)}\right]=e^{(4 x+5)} \cdot \frac{d}{d x}(4 x+5) \\
& =e^{(4 x+5)} \times(4 \times 1+0)=4 e^{(4 x+5)}
\end{aligned}
$
$
\begin{aligned}
& \text { and } \begin{aligned}
\frac{d v}{d x} & =\frac{d}{d x}\left(10^{4 x}\right)=10^{4 x} \cdot \log 10 \cdot \frac{d}{d x}(4 x) \\
& =10^{4 x} \cdot(\log 10) \times 4=4 \cdot 10^{4 x} \cdot \log 10 \\
\therefore \frac{d u}{d v} & =\frac{(d u / d x)}{(d v / d x)}=\frac{4 e^{(4 x+5)}}{4 \cdot 10^{4 x} \cdot \log 10}=\frac{e^{(4 x+5)}}{10^{4 x} \cdot \log 10} .
\end{aligned}
\end{aligned}
$

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