$$\Pr(X\le x) = 0.6 \Phi(x) + 0.4 \Phi\left( \frac x {\sqrt 2} \right)$$$$\frac d {dx} \Pr(X\le x) = 0.6 \varphi(x) + 0.4 \varphi\left( \frac x {\sqrt 2} \right) \cdot \frac 1 {\sqrt 2}$$(The chain rule was used in the second term.)
Recall that $\displaystyle \varphi(x) = \frac 1 {\sqrt{2\pi}} e^{-x^2/2}$.
