# Intuitive meaning of second, third and fourth derivatives at a point.

Can someone explain me the intuitive meaning of second, third and fourth derivatives of a function say, $f(x)$ at a point (say, $a$)? I know it's hard to explain to someone novice like me! But an intuitive answer of this question can help many people.

To remain intuitive, the Taylor expansion of a function around a point $x_0$ is nothing else than a polynomial of grade - let's say - $n$ that has as coefficients the various derivatives of the function at that point.
Of course, the $k$-th derivative of the function and of the polynomial will coincide at $x_0$. Thus, the $k$-th derivative of a function has the meaning of the coefficient of $x^k$ of the polynomial given by the taylor expansion, which is tangent at the function in $x_0$. Also more explicitly, the equation $$p_k(x) - f(x) = 0$$ will have a root of multiplicity $k$ at $x_0$ if $p_k(x)$ is the Taylor expansion to order $k$ of $f(x)$. For $k=2$ this corresponds to the curvature of the circle tangent to the function in $x_0$ called "osculator circle".