Find d y d x if, : y=(2 x+5)^x - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=(2 x+5)^x
$

✓

Answer

$
\begin{aligned}
& y=(2 x+5)^x \\
& \therefore \log y=\log (2 x+5)^x=x \log (2 x+5)
\end{aligned}
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
\frac{1}{y} \cdot \frac{d y}{d x} & =\frac{d}{d x}[x \log (2 x+5)] \\
& =x \frac{d}{d x}[\log (2 x+5)]+[\log (2 x+5)] \cdot \frac{d}{d x}(x) \\
& =x \times \frac{1}{2 x+5} \cdot \frac{d}{d x}(2 x+5)+[\log (2 x+5)] \times 1 \\
& =\frac{x }{2 x+5} \times(2 \times 1+0)+\log (2 x+5) \\
\therefore \frac{d y}{d x} & =y\left[\frac{2 x}{2 x+5}+\log (2 x+5)\right] \\
& =(2 x+5)^x\left[\log (2 x+5)+\frac{2 x}{2 x+5}\right] .
\end{aligned}
$

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