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I recently noticed that

$\tan{\theta} = \frac{dy}{dx}$

where $\theta$ is the angle of the tangent line to $y(x)$ with respect to the x-axis. Does this relationship have any practical uses, i.e. as a shortcut to finding derivatives, and if not, is there any context this would be useful?

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It is true for linear functions in general. If you have the line $y=mx+b$ the tangent of the angle from the $x$ axis to the line has $\tan \theta=m$. The tangent is just one more straight line.

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    For a linear function, isn't the tangent line at all points identical to line itself? I do the think that is generalizing the angle-derivative relationship, it looks to me like that is restricting it to a specific type of function. – 2017-01-24
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    Yes, the tangent line of a linear function is identical to the line itself. My point is that the relationship between the slope of the line and the angle is independent of how you got the line. The fact that the line is a tangent does not matter. – 2017-01-24