Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$3^{x^{2}+2 x}$

Solution:

Let $y=3^{x^{2}+2 x}$

On differentiating $y$ with respect to $x$, we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(3^{\mathrm{x}^{2}+2 \mathrm{x}}\right)$

We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{a}^{\mathrm{x}}\right)=\mathrm{a}^{\mathrm{x}} \log \mathrm{a}$

$\Rightarrow \frac{d y}{d x}=3^{x^{2}+2 x} \log 3 \frac{d}{d x}\left(x^{2}+2 x\right)$ [using chain rule]

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{x}^{2}+2 \mathrm{x}} \log 3\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)+\frac{\mathrm{d}}{\mathrm{dx}}(2 \mathrm{x})\right]$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{x}^{2}+2 \mathrm{x}} \log 3\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)+2 \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})\right]$

We have $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ and $\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})=1$

$\Rightarrow \frac{d y}{d x}=3^{x^{2}+2 x} \log 3[2 x+2 \times 1]$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{x}^{2}+2 \mathrm{x}} \log 3(2 \mathrm{x}+2)$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=(2 \mathrm{x}+2) 3^{\mathrm{x}^{2}+2 \mathrm{x}} \log 3$

Thus, $\frac{d}{d x}\left(3^{x^{2}+2 x}\right)=(2 x+2) 3^{x^{2}+2 x} \log 3$