In the previous section we learned how to calculate the derivative $f^{\prime}(x)$ of any function $f(x)$. The second derivative of $f(x)$ is the derivative of the derivative of $f(x)$ and it is denoted as: \[ f^{\prime\prime}(x) \equiv \left[ f^{\prime}(x) \right]^\prime \equiv \frac{d}{dx} f^{\prime}(x). \equiv \frac{d^2}{dx^2} f(x). \] This process can be continued further in order to calculate higher derivatives of $f(x)$.

In practice, the first and second derivatives are most important because they have a geometrical interpretation. The first derivative of $f(x)$ describes the slope of $f(x)$ while the second derivative describes the curvature of $f(x)$.

• $f(x)$: A function.
• $f^{\prime}(x)$: The first derivative of the function $f(x)$.

The first derivatives contains the information about the slope of the function.

• $f^{\prime\prime}(x)$: The second derivative of the function $f(x)$.

The second derivative contains the information about the curvature of the function $f(x)$.

• If $f^{\prime\prime}(x)>0$, then the function $f(x)$ is convex (opens upwards).
• If $f^{\prime\prime}(x)<0$, then the function $f(x)$ is concave (opens downwards).
• $f^{\prime\prime\prime}(x)\equiv f^{(3)}(x)$: The third derivative of the function $f(x)$.
• $f^{(n)}(x)$: The n$^\textrm{th}$ derivative of the function $f(x)$.

The second derivative describes the change in the value of the first derivative. To obtain $f^{\prime\prime}(x)$ we compute the derivative of $f'(x)$.

The second derivative tells you about the “curvature” of the function $f(x)$. If the curvature of a function is positive ($f^{\prime\prime}(x)>0$), this means that the slope of the function is increasing so the function must be curving upwards. Negative curvature means the function curves downwards.

Calculate the second derivatives of the functions $u(x)=x^2$ and $d(x)=-x^2$ and comment on the shape of these functions.

To solve this problem, we first calculate the first derivative $u^{\prime}(x)=2x$ and $d^{\prime}(x)=-2x$. We obtain the second derivatives by taking the derivative of the first derivative: $u^{\prime\prime}(x)=2$ and $d^{\prime\prime}(x)=-2$. The fact that the second derivative is positive means that the curvature of the function $u(x)$ is always positive. The function $u(x)$ is convex: it opens upwards. On the other hand $d(x)$ is concave: it opens downwards.

The function $u(x)$ and $d(x)$ are canonical examples of functions with positive and negative curvature. If a function $f(x)$ has positive curvature at a point $x^*$ ($f^{\prime\prime}(x^*) > 0$), then the function locally resembles $u(x-x^*)=(x-x^*)^2$. If on the other hand the second derivative of $f(x)$ is negative at $x^*$, then the function locally resembles $d(x-x^*)=-(x-x^*)^2$. In other words, the terms convex and concave refer to the $u$-likeness vs. $d$-likeness property of functions.

If we take the derivative of the derivative of the derivative of $f(x)$ we obtain the third derivative of the function. This process can be continued further to obtain the n$^\textrm{th}$ derivative of the function: \[ f^{(n)}(x) \equiv \frac{d^n}{dx^n} f(x) \equiv \underbrace{ \frac{d}{dx} \frac{d}{dx} \cdots \frac{d}{dx} }_{n} f(x). \]

Higher derivatives do not have an obvious geometrical interpretation. However, if you are given a function $f(x)$ such that $f^{\prime\prime\prime}(x)>0$, then the function $f(x)$ must be $+x^3$-like. Alternately, if $f^{\prime\prime\prime}(x)<0$, then the function must resemble $-x^3$.

Later in this chapter, we will learn how to compute the Taylor series of a function, which is a procedure used to find polynomial approximations to any function $f(x)$: \[ f(x) \ \approx \ a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + \cdots + a_n x^n. \] The values of the coefficients $a_0$, $a_1$, $\ldots$, $a_n$ in the approximation will require computing higher derivatives of $f(x)$. The coefficient $a_n$ tells us whether $f(x)$ is more similar to $+x^n$ or $-x^n$.

Compute the third derivative of $f(x)=\sin(x)$.

The first derivative is $f^{\prime}(x)=\cos(x)$. The second derivative will be $f^{\prime\prime}(x)=-\sin(x)$ and so the third derivative must be $f^{\prime\prime\prime}(x)=-\cos(x)$.