7.3 Higher Derivatives

If the derivative of a differentiable function can be considered a function in its own right, then we can investigate the differentiability of it as well. If the derivative $f^{\prime}$ of a function $f$ is differentiable, then we can define another function $f^{\prime\prime}$ or $\frac{\mathrm{d}^{2}f}{\mathrm{d}x^{2}}$ as the second derivative, and we say that $f$ is twice differentiable. We can carry on this way if the derivatives carry on being differentiable, and once the prime notation becomes unwieldy, we can write the $n$ ^th derivative of $f$ as $f^{(n)}$ or $\frac{\mathrm{d}^{n}f}{\mathrm{d}x^{n}}$ .

7.3.1 Operations on Derivatives

Theorem 7.6

Let $I\subseteq\mathbb{R}$ and let $f,g:I\to\mathbb{R}$ be two functions which are twice differentiable at some point $c\in I$ .

(i)

$(f+g)^{\prime\prime}(c)=f^{\prime\prime}(c)+g^{\prime\prime}(c)$ .
(ii)

Given $\lambda\in\mathbb{R}$ , $(\lambda f)^{\prime\prime}(c)=\lambda f^{\prime\prime}(c)$ .
(iii)

$(fg)^{\prime\prime}(c)=f^{\prime\prime}(c)g(c)+2f^{\prime}(c)g^{\prime}(c)+f(c% )g^{\prime\prime}(c)$ .

Proof.

(i)

By theorem 7.2 (i), $(f+g)^{\prime}(c)=f^{\prime}(c)+g^{\prime}(c)$ . Therefore,

$(f+g)^{\prime\prime}(c)=(f^{\prime}+g^{\prime})^{\prime}(c)=f^{\prime\prime}(c% )+g^{\prime\prime}(c).$ (7.39)
(ii)

By theorem 7.2 (ii), $(\lambda f)^{\prime}(c)=\lambda f^{\prime}(c)$ . So,

$(\lambda f)^{\prime\prime}(c)=(\lambda f^{\prime})^{\prime}(c)=\lambda f^{% \prime\prime}(c).$ (7.40)

(iii)

By theorem 7.2 (iii), $(fg)^{\prime}(c)=f(c)g^{\prime}(c)+f^{\prime}(c)g(c)$ . So,

$\displaystyle(fg)^{\prime\prime}(c)$	$\displaystyle=(f(c)g^{\prime}(c)+f^{\prime}(c)g(c))^{\prime}$	(7.41)
	$\displaystyle=(f(c)g^{\prime}(c))^{\prime}+(f^{\prime}(c)g(c))^{\prime}$	(7.42)
	$\displaystyle=(f(c)g^{\prime\prime}(c)+f^{\prime}(c)g^{\prime}(c))+(f^{\prime}% (c)g^{\prime}(c)+f^{\prime\prime}(c)g(c))$	(7.43)
	$\displaystyle=f^{\prime\prime}(c)g(c)+2f^{\prime}(c)g^{\prime}(c)+f(c)g^{% \prime\prime}(c).$	(7.44)

∎

The first two parts of this theorem generalise very easily by induction to all higher derivatives. The final part is more involved.

Theorem 7.7 (General Leibniz Rule)

Let $I\subseteq\mathbb{R}$ and let $f,g:I\to\mathbb{R}$ be two $n$ -times differentiable functions at some point $c\in I$ . Then the $n$ ^th derivative of the product $(fg):I\to\mathbb{R}$ at $c$ takes the value

(fg)^{(n)}(c)=\sum_{k=0}^{n}\binom{n}{k}f^{(n-k)}(c)g^{(k)}(c).

(7.45)

7.3.2 Taylor’s Theorem

The last theorem of this chapter is a generalisation of the Mean Value theorem. It allows us to approximate the value of a function at a point as a polynomial with the derivatives of the function as coefficients.

Theorem 7.8 (Taylor’s Theorem)

Let $I$ be some open interval and let $a,b\in I$ . Let $f:I\to\mathbb{R}$ be a continuous function which is $(n+1)$ -times differentiable at $a$ . Then there exists $\theta$ between $a$ and $b$ such that

f(b)=f(a)+(b-a)f^{\prime}(a)+\frac{(b-a)^{2}}{2!}f^{\prime\prime}(a)+\dots+% \frac{(b-a)^{n}}{n!}f^{(n)}(a)+\frac{(b-a)^{n+1}}{(n+1)!}f^{(n+1)}(\theta).

(7.46)

Proof.

First we define $F:[a,b]\to\mathbb{R}$ to be given by

F(t)=f(b)-f(t)-(b-t)f^{\prime}(t)-\frac{(b-t)^{2}}{2!}f^{\prime\prime}(t)-% \dots-\frac{(b-t)^{n}}{n!}f^{(n)}(t).

(7.47)

Since the first $n$ derivatives of $f$ are continuous on $[a,b]$ and differentiable on $(a,b)$ , $F$ inherits these properties by theorem 7.2. The derivative of $F$ is then

$\displaystyle F^{\prime}(t)$	$\displaystyle=-f^{\prime}(t)+[f^{\prime}(t)-(b-t)f^{\prime\prime}(t)]+\left[(b% -t)f^{\prime\prime}(t)-\frac{(b-t)^{2}}{2!}f^{\prime\prime\prime}(t)\right]$	(7.48)
	$\displaystyle\quad\,+\dots+\left[\frac{(b-t)^{n-1}}{(n-1)!}f^{(n)}(t)-\frac{(b% -t)^{n}}{n!}f^{(n+1)}(t)\right]$	(7.49)
	$\displaystyle=-\frac{(b-t)^{n}}{n!}f^{(n+1)}(t).$	(7.50)

Now, define $G:[a,b]\to\mathbb{R}$ to be given by

G(t)=F(t)-\left(\frac{b-t}{b-a}\right)^{n+1}F(a).

(7.51)

Notice that $G(a)=G(b)=0$ , so by Rolle’s theorem (7.4) there exists $\theta\in(a,b)$ such that $G^{\prime}(\theta)=0$ . Specifically,

$\displaystyle G^{\prime}(\theta)$	$\displaystyle=F^{\prime}(\theta)+(n+1)\frac{(b-\theta)^{n}}{(b-a)^{n+1}}F(a)$	(7.52)
	$\displaystyle=-\frac{(b-\theta)^{n}}{n!}f^{(n+1)}(\theta)+(n+1)\frac{(b-\theta% )^{n}}{(b-a)^{n+1}}F(a)$	(7.53)
	$\displaystyle=(n+1)\frac{(b-\theta)^{n}}{(b-a)^{n+1}}\left[F(a)-\frac{(b-a)^{n% +1}}{(n+1)!}f^{(n+1)}(\theta)\right]=0.$	(7.54)

The prefactor is never zero, hence the term in square brackets must be zero, so

F(a)=\frac{(b-a)^{n+1}}{(n+1)!}f^{(n+1)}(\theta),

(7.55)

	$\displaystyle\implies f(b)=$	$\displaystyle f(a)+(b-a)f^{\prime}(a)+\frac{(b-a)^{2}}{2!}f^{\prime\prime}(a)$		(7.56)
		$\displaystyle+\dots+\frac{(b-a)^{n}}{n!}f^{(n)}(a)+\frac{(b-a)^{n+1}}{(n+1)!}f% ^{(n+1)}(\theta).$		(7.57)

∎

7.3.3 Approximating Functions

Taylor’s theorem allows us to make approximations that are much better than linear ones. If, like before, we replace $b$ any point $x$ in the interior of the domain of $f$ , we get

	$\displaystyle f(x)$	$\displaystyle=f(a)+(x-a)f^{\prime}(a)+\frac{(x-a)^{2}}{2!}f^{\prime\prime}(a)$		(7.58)
		$\displaystyle+\dots+\frac{(x-a)^{n}}{n!}f^{(n)}(a)+\frac{(x-a)^{n+1}}{(n+1)!}f% ^{(n+1)}(\theta),$		(7.59)

where $\theta$ is now between $x$ and $a$ . This polynomial representation of $f$ is known as the $n$ ^th Taylor Polynomial of $f$ about $a$ and the last term is known as the error or remainder, sometimes denoted $R_{n}(x)$ . The idea is that for well-behaved functions, the remainder will get smaller as $n$ gets larger.

Corollary 7.8.1 (Maclaurin’s Theorem)

Suppose $0\in I$ . Let $x\in I$ , then if $f:I\to\mathbb{R}$ is an $(n+1)$ -times differentiable function at 0, then there exists $\theta$ between $x$ and 0 such that

f(x)=f(0)+xf^{\prime}(0)+\frac{x^{2}}{2!}f^{\prime\prime}(0)+\dots+\frac{x^{n}% }{n!}f^{(n)}(0)+R_{n}(x),

(7.60)

where $R_{n}(x)=\frac{x^{n+1}}{(n+1)!}f^{(n+1)}(\theta)$ .

Proof.

This is easily shown by using Taylor’s theorem with $a=0$ . ∎

Taylor’s theorem and Maclaurin’s theorem have very important consequences that we will explore in more detail in later chapters.