Is this right? I differentiated $P$ to find $\dfrac{\mathrm dP}{\mathrm dQ}$ then put $ \dfrac{1}{\mathrm dP/\mathrm dQ} $ to find $\dfrac{\mathrm dQ}{\mathrm dP}$ and multiplied $\dfrac{\mathrm dQ}{\mathrm dP}$ by $\dfrac{P}{Q}$ to get the PED expression in terms of $Q$.
The demand is given by the equation:
$$P = 920Q^{-0.4}e^{-0.00005Q}.$$
My answer: $$ \left(-\frac{x^{1.4}}{368} \cdot (-20000)x^{0.00005}\right) (920Q^{-0.4}e{^{0.00005Q}}).$$
Is my thought process and answer correct?
The demand equation implicitly defines $Q(P)$, so define
$$ F(P,Q) \equiv P - 920Q^{-0.4}e^{-0.00005Q} =0 $$
and use the implicit function theorem to get
$$ \frac{dQ}{dP}= -\frac{F_P(P,Q)}{F_Q(P,Q)}=-\frac{1}{920(0.4+0.00005Q)Q^{-1.4}e^{-0.00005Q}} = -\frac{Q^{1.4}e^{0.00005Q}}{920(0.4+0.00005Q)}.$$
Then, it follows that the price elasticity of demand is
$$ \frac{dQ}{dP}\frac{P}{Q}=-\frac{Q^{1.4}e^{0.00005Q}}{920(0.4+0.00005Q)}\frac{920Q^{-0.4}e^{-0.00005Q}}{Q}= -\frac{1}{0.4+0.00005Q}.$$