Question
Diffrentiate the following w.r.t.x

$\sqrt{e^{(3 x+2)}+5}$

Answer

Let $y =\sqrt{e^{(3 x+2)}+5}$

Differentiating w.r.t. x, we get

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[e^{(3 x+2)}+5\right]^{\frac{1}{2}} \\ & =\frac{1}{2}\left[e^{(3 x+2)}+5\right]^{-\frac{1}{2}} \cdot \frac{d}{d x}\left[e^{(3 x+2)}+5\right] \\ & =\frac{1}{2 \sqrt{e^{(3 x+2)}+5}} \cdot\left[e^{(3 x+2)} \cdot \frac{d}{d x}(3 x+2)+0\right] \\ & =\frac{1}{2 \sqrt{e^{(3 x+2)}+5}} \cdot\left[e^{(3 x+2)} \cdot(3 \times 1+0)\right] \\ & =\frac{3 e^{(3 x+2)}}{2 \sqrt{e^{(3 x+2)}+5}} \cdot\end{aligned}$

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