Differentiate the following w.r.t. x. : x^3 3^x - Vidyadip

Question

Differentiate the following w.r.t. x. : \begin{equation}
x^3 \cdot 3^x
\end{equation}

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Answer

Let $y=x^3 \cdot 3^x$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(x^3 3^x\right) \\
& =x^3 \frac{\mathrm{d}}{\mathrm{d} x}\left(3^x\right)+3^x \frac{\mathrm{d}}{\mathrm{d} x}\left(x^3\right) \\
& =\left(x^3\right)\left(3^x \log 3\right)+3^x\left(3 x^2\right) \\
& =x^2 3^x(x \log 3+3)
\end{aligned}
$

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