Differentiating $\frac{1}{Q(x)}$ and $\frac{1}{\sqrt{2 \pi}}e^{-\frac{x^2}{2 }}$

Question

I am not very sure about differentiating $\frac{1}{Q(x)}$ and $\frac{1}{\sqrt{2 \pi}}e^{-{x^2}/{2}}$

$$Q(x)=\frac{1}{\sqrt{2 \pi}}\int ^{\infty}_{x} e^{-{t^2}/{2}} \mathrm dt$$

Are my calculations below correct?

$$\left(\frac{1}{Q(x)}\right)'= \frac{1}{Q(x)^2}$$ $$\left(\frac{1}{\sqrt{2 \pi}}e^{-{x^2}/2}\right)'=\frac{x}{\sqrt{2 \pi}}e^{-{x^2}/{2}}$$

Answer 1

$$\left( \frac1{Q(x)}\right)'=\left((Q(x))^{-1}\right)'=-Q(x)^{-2}Q'(x)$$

Try to use fundamental theorem of calculus.

$$\left( \frac1{\sqrt{2\pi}} \exp\left( -\frac{x^2}{2}\right)\right)'=\frac1{\sqrt{2\pi}} \exp\left( -\frac{x^2}{2}\right)\frac{d}{dx}\left( -\frac{x^2}{2}\right)$$

Try to differentiate again.