How to calculate $\frac{d}{d\lambda}(f(\lambda x,\lambda y))$

Question

I'm solving a problem from my differential equation's course, and for certain prove I need to calculate $$\frac{d}{d\lambda}g(\lambda),$$ being $g(\lambda)=f(\lambda x,\lambda y)$. A classmate sent me a photo of his solution and I noticed he stated that $$\frac{d}{d\lambda}(f(\lambda x,\lambda y))=\frac{\partial f(\lambda x,\lambda y)}{\partial \lambda x}\cdot\frac{\partial\lambda x}{\partial\lambda} + \frac{\partial f(\lambda x,\lambda y)}{\partial \lambda y}\cdot\frac{\partial\lambda y}{\partial\lambda}.$$ Is this true? If so, why is it true? I'm having trouble figuring out how this derivative works. Any help will be appreciated.