X. Differentiation #

Problems on the derivative: its consequences for monotonicity, the existence of higher derivatives, the fact that a derivative need not be continuous, and a construction of the power rule.

Problem 1 #

Suppose that $f$ is differentiable at every point of $[a,b]$ and suppose that the derivative is never zero. Prove that $f$ is strictly monotonic on $[a,b]$. Note that $f'$ is not assumed to be continuous.

Solutions: Hint · Solution · Long solution

Problem 2 #

Does the derivative of a function need to be a continuous function? Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by

$$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0, \\[6pt] 0 & \text{if } x = 0. \end{cases}$$

Show that its derivative is not continuous. To make life easier, you may assume the following:

(a) $\sin(x)$ and $\cos(x)$ are continuous and differentiable for all values of $x$;

(b) $\sin'(x) = \cos(x)$ and $\cos'(x) = -\sin(x)$;

(c) $\tfrac{1}{x}$ is differentiable on the open set $\mathbb{R}\setminus \{0\}$ and $\left(\tfrac{1}{x}\right)' = -\tfrac{1}{x^2}$;

(d) the limit $\lim_{x \to 0} \cos\left(\tfrac{1}{x}\right)$ does not exist; and

(e) if $\lim_{x\to c} f(x)$ exists and $\lim_{x\to c} g(x)$ does not, then $\lim_{x\to c} f(x) + g(x)$ does not exist.

Hint: First, compute a formula for $f'(x)$. The expression will depend on whether or not $x = 0$. When $x\neq 0$, the usual calculus formulas apply. When $x=0$, use the limit definition of a derivative to compute $f'(0)$.

Solutions: Hint · Solution · Long solution

Problem 3 #

Find a function whose second derivative $f''(c)$ exists at some point $c$, but $f'''(c)$ does not.

Solutions: Hint · Solution · Long solution

Problem 4 #

The goal of this sequence of exercises is to derive the power rule for differentiation. It begins with a formal definition of the natural logarithm.

(a) Define a differentiable function $L: \mathbb{R}_{>0} \to \mathbb{R}$ by requiring that

(i) $L'(x) = \frac{1}{x}$, and

(ii) $L(1)=0$.

Prove that these conditions define $L$ uniquely. That is, if $M$ is another function satisfying both of the above, then $L(x)=M(x)$ for all $x \in \mathbb{R}_{>0}$. We will write $\ln(x)$ instead of $L(x)$.

(b) Show that $\ln x$ is a bijection from $\mathbb{R}_{>0}$ to $\mathbb{R}$. Hint: To show that it is surjective, first show that it has neither an upper nor a lower bound and then use the intermediate value theorem. One way to show that $\ln x$ fails to have an upper bound is to first show that $\ln n \geq \sum_{k=2}^n \frac{1}{k}$ for all $n$.

(c) Since $\ln x$ is a bijection, it has an inverse function, which we define to be $e^x$. Using the chain rule, prove that $(e^x)'=e^x$.

(d) For a positive real number $x$ and any real $\alpha$, we can now define $x^\alpha$ as

$$x^\alpha = e^{\alpha \ln x}.$$

Armed with this definition, show that

$$(x^\alpha)' = \alpha x^{\alpha -1}.$$

Solutions: Hint · Solution · Long solution