# Why derivative is a slope?

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.

• If the graph is curved, then the slope can be different at every point. This is why it still depends on $x$. – Nick Feb 17 at 0:41
• 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. – Ian Feb 17 at 0:43
• @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? – Anon Alexander Feb 17 at 0:53
• 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 $(x,f(x))$. – littleO Feb 17 at 1:06
• 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. – Filip Milovanović Feb 17 at 9:29

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 $$x$$.
• 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 – whpowell96 Feb 17 at 1:09
• 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. – Sam Feb 17 at 4:24
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.