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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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    $\begingroup$ If the graph is curved, then the slope can be different at every point. This is why it still depends on $x$. $\endgroup$ – Nick Feb 17 at 0:41
  • $\begingroup$ The derivative at a particular point is a number which gives the slope of the tangent line at that particular point. For example, the tangent line of $y=3x^2$ at $x=1$ is the line $y=6(x-1)+3$. But the slope of the tangent line is generally not the same at each point. $\endgroup$ – Ian Feb 17 at 0:43
  • $\begingroup$ @lan so $3x^2$ is a function, and the derivative of this function is $6x$. Can we say this $6x$ is also a function, and the derivative of this function is 6? Or "the derivative of $3x^2$ is $6x$" that's the end of the story? $\endgroup$ – Anon Alexander Feb 17 at 0:53
  • $\begingroup$ My attempt at clarifying terminology: Suppose $f:\mathbb R \to \mathbb R$ is the function defined by $f(x) = x^3$ for all $x \in \mathbb R$. Then $f'$ is a function, and if $x \in \mathbb R$ then $f'(x) = 3x^2$. The function $f'$ is called the derivative of $f$. If $x \in \mathbb R$, the number $f'(x)$ is called the derivative of $f$ at $x$. So $f'$ is a function, but $f'(x)$ is a number. The number $f'(x)$ is the slope of the tangent line to the graph of $f$ at the point $(x,f(x))$. $\endgroup$ – littleO Feb 17 at 1:06
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    $\begingroup$ So when people say that the derivative is a slope, they either mean that it's a function that tells you the slope, or that a derivative evaluated at a certain point is a slope. $\endgroup$ – Filip Milovanović Feb 17 at 9:29
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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 $x$.

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    $\begingroup$ So a derivative is not a slope. It's a function of another function that calculates the slope at each point. So it is not a slope of a slope? $\endgroup$ – Anon Alexander Feb 17 at 0:57
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    $\begingroup$ Yes that is one way to put it. I wouldn't exactly call it "a function of another function" but instead I would consider $f'$ to be a function derived from the function $f$, but that's just semantics $\endgroup$ – whpowell96 Feb 17 at 1:09
  • $\begingroup$ The slope of the slope or $f''(x)$ represents the rate at which slope is increasing. From, here we can infer that if $f'(x)=0$ and $f''(x)>0$, then $x$ is a point of minima. If $f''(x)<0$, then the point is the maxima. $\endgroup$ – Sam Feb 17 at 4:24
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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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