The change of $Y$ per $X$ is slope. And some say the change of slope per $X$ is derivative. So it is like slope of a slope!
But slopes are always numbers like the slope of $2x$ is $2$. But derivates are not just numbers like the derivative of $3x^2$ is $6x$. This is confusing me can someone please explain? Thank you.
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$\begingroup$The derivative of $f$ at the point $x$ is the slope of the tangent line to $f$ at the point $x$. So for $f(x) = 3x^2$, we have $f'(x) = 6x$. What this means is that the derivative function $f'(x)$ takes in a value $x$ and returns the slope of the tangent line of $f$ at the point $x$. You can view the derivative function as a function that takes in points and returns tangent slopes to $f$ at said points. Each of these slopes is simply a number, but of course the tangent slope depends on what point you are computing the tangent at, hence the dependence of $f'$ on $x$.
$\endgroup$ 3 $\begingroup$If you were to draw a graph of $3x^2$, you would notice that the slope is different for every value of $x$, so the derivative of the derivative is basically, like you put it, in some ways the slope of a slope.
It might make it easier to not see it as a slope, though, but more as an increment of $y$ per increment of $x$. You can see that that changes throughout the graph because at some points the graph is “steeper”, implying that a small step taken on the $x$-axis could result in a huge step on the $y$-axis. So the derivative of the derivative boils down to how much steeper the graph is getting at certain points.
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