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How can I explain to someone who is a beginner, the basic idea and intuition behind dy/dx? I'm introducing calculus to one of my friends and here are the questions I might encounter: Why dy/dx? What do I even need to find the slope? How does that relate to rate of change of y with respect to x? What could be some real life examples for a better understanding? What information does the slope at every point of a curve, which is just a line, give me?

I'd like to explain all of this giving an intuitive sense to the formal differential calculus. Thanks!

user231094
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    Remember that you are using cartesian coordinates to plot the function $f(x)$. Thus, what you are "measuring" is the rate of change of function $f$, i.e. $\dfrac {f(x+dx)-f(x)}{dx}$. – Mauro ALLEGRANZA May 13 '19 at 13:32
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    If you have an afternoon, you could take a look at these videos. – Arthur May 13 '19 at 13:35
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    These are all valid questions but not for someone who is attempting to explain Calculus. Your friend would be better served by someone who has mastered the basics. – John Douma May 13 '19 at 13:35
  • start with linear functions, consider $y=x$, $y=2x$, $y=-x$, $y=2$. Show what happens to value of $y$ if $x$ is increased by $dx$. – Vasili May 13 '19 at 13:38
  • It's the limit of $\frac{\Delta y}{\Delta x}$ as $\Delta x\to 0.$ – Thomas Andrews May 13 '19 at 13:43

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