TITLE: What does the derivative of area with respect to length signify?
QUESTION [9 upvotes]: Suppose that we have a square sheet of edge length $L$. Its area $A=L^2$.  
Differentiating $A$ w.r.t. L, we get
$$\dfrac{dA}{dL}=2L$$  

I do understand what it means to differentiate, graphically, it gives you the slope of the tangent at a point on the graph. But now, when I think of what differentiating means in the context of Area and length, it doesn't make any sense to me at all. What does $2L$ signify?

REPLY [10 votes]: Consider this picture:

Here, the green square is the square of area $A=L^2$ and red line is its increase.
When you increase the length $L$ by $dL$, the area $A$ gets increased by $2LdL$. So, to answer your question, significance of $2L$ is that it is the length of the red line on the picture ($dL$ is its width).