$\begin{align}  & y=u+v \\ & \text{Where }y,u,v\text{ are functionof }x\text{ increase by }\delta y,\delta u, \\ & \delta v\text{ respectively as }x\text{ increase by a small amount } \\ & \text{of }\delta x \\ & \text{Then }y+\delta y=(u+\delta u)+(v+\delta v) \\ & \delta y=(u+\delta u)+(v+\delta v)-(u+v) \\ & \delta y=\delta u+\delta v \\ & \text{Dividing through by }\delta x \\ & \frac{\delta y}{\delta x}=\frac{\delta u}{\delta x}+\frac{\delta v}{\delta x} \\ & \text{Taking limit as }\delta x\to 0 \\ & \frac{dy}{dx}=\underset{\delta x\to 0}{\mathop{\lim }}\,\left( \frac{\delta u}{\delta x}+\frac{\delta v}{\delta x} \right) \\ & \frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx} \\\end{align}$