Find d y d x if, : Differentiate 5^x with respect to x. - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
Differentiate $5^x$ with respect to $\log x$.

✓

Answer

Let $u=5^x$ and $v=\log x$
Then we want to find $\frac{d u}{d v}$
Differentiating $u$ and $v$ w.r.t. $x$, we get
$
\begin{gathered}
\frac{d u}{d x}=\frac{d}{d x}\left(5^x\right)=5^x \cdot \log 5 \\
\text { and } \frac{d v}{d x}=\frac{d}{d x}(\log x)=\frac{1}{x} \\
\therefore \frac{d u}{d v}=\frac{(d u / d x)}{(d v / d x)}=\frac{5^x \cdot \log 5}{\left(\frac{1}{x}\right)}
\end{gathered}
$
$
=x \cdot 5^x \cdot \log 5 \text {. }
$

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