Whenever we want to find the slope of a graph whose equation is given, we find the arctan of dy/dx after substitution. But what makes us do this? Can this be proved?
Any help is appreciated.
Why should dy/dx always be the tangent of the inclination?
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calculus
graphing-functions
tangent-line
slope
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1$dy/dx$ is how much the function $y(x)$ increases for a tiny increment of $x$. Draw it like a triangle and there you'll have it. – 2017-02-08
1 Answers
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$\frac{dy}{dx}$ is, by definition, the limit of a secant line as the distance between two points approaches zero - it simply is the slope, nothing more to prove really (other than that the derivative actually exists, which is beyond the scope of this question)
As for the arctangent, it comes from the fact that $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ for right triangles; since we define slope as $\frac{\Delta y}{\Delta x}$ and this is basically what the derivative gives you (a small change in $y$ over a small change in $x$), we can find the inclination $\theta$ by taking the arctangent of both sides.