Why does $\frac{dx^\mu}{d\lambda}(\frac{\partial}{\partial{x^\mu}}\frac{dx^\nu}{d\lambda})=\frac{d^2x^\nu}{d\lambda^2}$?
I have got as far as using the product rule:
($\frac{dx^\mu}{d\lambda}\frac{\partial}{\partial{x^\mu}})\frac{dx^\nu}{d\lambda}+\frac{\partial}{\partial{x^\mu}}(\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda})$
but don't know how to evaluate the mixing of partial and total derivatives.