The $ nth $ derivative of a function tells the rate of change of the $ (n-1)th $ derivative of a function, and if $n=1$, we'll get the first derivative of our function.
So consider a function
$y = x^2$
$\implies$ $\frac {d(x^2)}{dx} = 2x$
This means that the slope of the tangent line at any $x$ value of the graph is $2x$.
But what happens when we further differentiate our first derivative?
We have $\frac{d(2x)}{dx} = 2 $
So if we plot $y'(x)$ on the graph, we get to know that the slope of $y'(x)$ at any $x$ value is 2. This means that the graph of $y'(x)$ is a straight line. So my question is :
Is it true that every differentiable function's last derivative expression, when plotted, will be a straight line?
Or in other words, can every differentiable function be differentiated till we get a constant?
EDIT : Every differentiable function can be differentiated infinitely many times, and the derivative of $0$ is $0$. I thought that $0$ couldn't be differentiated further.
Consider $f(x) = e^x$. Every derivative of $f$ is also $e^x$, so it will never become constant. In fact, almost all functions have the property that no amount of differentiating will produce a constant.
It is true, however, that for any polynomial $g(x)$ of degree $n$, the $n$th derivative of $g$ is constant (and so the $n+1$-st derivative is the zero function).