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I'm starting to learn calculus, and I'm getting confused about what average slope and instant slope(instantaneous rate of change)do and what they're differences are after looking at several sources on the internet. I know that average slope is $\frac{Δy}{Δx }$ and instant slope is $\frac{dy}{dx}$. Are these formulas correct? And if they are, what difference is there between Δy and dy?

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  • $\begingroup$ Welcome to Maths SX! $\frac{\mathrm dy}{\mathrm dx}$ is the limit of $\frac{\Delta y}{\Delta x}$ as $\Delta x$ tends to $0$. Geometrically, the difference is the difference between the slope of the tangent line at a point and the slope of a chord of the curve passing through this point. $\endgroup$ – Bernard Jul 9 at 13:26
  • $\begingroup$ I'm not sure I understand that well, its sort of hard for me to take in $\endgroup$ – Heech.p Jul 9 at 13:27
  • $\begingroup$ What do you not understand well? $\endgroup$ – Bernard Jul 9 at 13:28
  • $\begingroup$ Think of $\dfrac{dy}{dx}$ as what you get by making $\Delta x$ smaller and smaller while computing $\dfrac{\Delta y}{\Delta x}$. $\endgroup$ – Yves Daoust Jul 9 at 13:30
  • $\begingroup$ From what I heard, $\frac{dy}{dx}$ in this fraction, dy and dx represent one number each, where $\frac{Δy}{Δx}$, Δy Δx, delta represents the change in the fucntion, is this correct? @Bernard $\endgroup$ – Heech.p Jul 9 at 13:35
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$\Delta y$ and $\Delta x$ represent actual numbers. If you have two points on the graph of a function, then $\Delta y$ is the change in their $y$-coordinates, and $\Delta x$ is the change in their $x$-coordinates. So, when you divide change in $y$ by change in $x$, i.e. $\frac{\Delta y}{\Delta x}$, you get the slope of the line that connects them. As you move the points closer together (i.e. make $\Delta x$ smaller and smaller), the line no longer connects two points, but becomes a line tangent to the graph of the function. The slope of that line is written as $\frac{dy}{dx}$, where here we think of $\frac{dy}{dx}$ as a single symbol, and not a fraction. Take a look at this Desmos link for a way to visualize this. The $h$ slider controls $\Delta x$, so as you make $h$ smaller and smaller you can see the slope $\frac{\Delta y}{\Delta x}$ get closer to 2, which is $\frac{dy}{dx}$, the slope of the tangent line at $x = 1$.

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    $\begingroup$ Thank you so much! $\endgroup$ – Heech.p Jul 9 at 15:16

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