Chapter 4 - Differentiable Functions, The Slope of a Curve, Rules for Differentiation, Mean Value Theorem and Beyond

4.1 The Derivative: Definition and Elementary Consequences - Zooming in on Change

We now come to the heart of differential calculus: the derivative. The derivative captures the instantaneous rate of change of a function at a point. Geometrically, it represents the slope of the tangent line to the graph of the function at that point.

Imagine zooming in on the graph of a function at a particular point $x_0$. If the function is “smooth” at that point, as you zoom in closer and closer, the graph will start to look more and more like a straight line. The slope of this line is the derivative at $x_0$.

To make this precise, we define the derivative using a limit:

Definition 4.1.1: Differentiability and the Derivative

Let $D\subseteq \mathbb{R}$, $f:D\to \mathbb{R}$, and $x_0\in D$ be a limit point of $D$. We say $f$ is differentiable at $x_0$ if the following limit exists:

$f^{\prime }(x_0)={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$

If this limit exists, we call it the derivative of $f$ at $x_0$, denoted by $f^{\prime }(x_0)$ or $\frac{df}{dx}(x_0)$.

The expression $\frac{f(x)-f(x_0)}{x-x_0}$ represents the slope of the secant line through the points $(x_0,f(x_0))$ and $(x,f(x))$. As $x$ approaches $x_0$, this secant line approaches the tangent line, and its slope approaches the derivative $f^{\prime }(x_0)$.

Alternative Form of the Derivative:

Sometimes it’s convenient to express the derivative using a change in $x$, denoted by $h=x-x_0$, so $x=x_0+h$. As $x\to x_0$, we have $h\to 0$. This gives us an equivalent definition:

Remark 4.1.2: Alternative Definition of the Derivative

$f^{\prime }(x_0)={\lim }_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$

This form is often useful in calculations and proofs.

Linear Approximation: Zooming in and Seeing a Line

Differentiability at a point $x_0$ means we can approximate the function near $x_0$ by a linear function. This is the essence of the “zooming in” idea.

Theorem 4.1.3: Linear Approximation Characterization of Differentiability

A function $f:D\to \mathbb{R}$ is differentiable at $x_0\in D$ if and only if there exists a real number $c$ and a function $r:D\to \mathbb{R}$ such that:

1. Linear Approximation Formula: For all $x\in D$, $f(x)=f(x_0)+c(x-x_0)+r(x)(x-x_0)$.
2. Remainder Term Vanishes Near $x_0$: ${\lim }_{x\to x_0}r(x)=0$, and we can define $r(x_0)=0$ to make $r$ continuous at $x_0$.

If these conditions hold, then $c$ is uniquely determined and $c=f^{\prime }(x_0)$.

This theorem provides a different perspective on differentiability. It says $f$ is differentiable at $x_0$ if we can write $f(x)$ as:

for $x$ close to $x_0$. The term $f(x_0)+f^{\prime }(x_0)(x-x_0)$ is the equation of the tangent line at $x_0$. The remainder term $r(x)(x-x_0)$ represents the “error” in this linear approximation, and it becomes negligible as $x$ approaches $x_0$ (since $r(x)\to 0$).

Proof of Theorem 4.1.3 (Sketch)

($\Rightarrow$) If $f$ is differentiable, then linear approximation exists: Define $c=f^{\prime }(x_0)$ and $r(x)=\frac{f(x)-f(x_0)}{x-x_0}-f^{\prime }(x_0)$ for $x\ne x_0$, and $r(x_0)=0$. Rearranging this gives the linear approximation formula. Taking the limit as $x\to x_0$ shows ${\lim }_{x\to x_0}r(x)=0$, so $r$ is continuous at $x_0$.

($\Leftarrow$) If linear approximation exists, then $f$ is differentiable: Given $f(x)=f(x_0)+c(x-x_0)+r(x)(x-x_0)$ with ${\lim }_{x\to x_0}r(x)=0$. Rearranging: $\frac{f(x)-f(x_0)}{x-x_0}=c+r(x)$. Taking the limit as $x\to x_0$: ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}={\lim }_{x\to x_0}(c+r(x))=c+0=c$. So the derivative exists and is equal to $c$.

This linear approximation view is fundamental for understanding applications of derivatives in physics, engineering, and other fields, where we often approximate nonlinear functions with linear ones locally.

Differentiability Implies Continuity: Smoothness Requires No Jumps

If a function is differentiable at a point, it must also be continuous at that point. Smoothness is a stronger condition than just being “connected”.

Corollary 4.1.5: Differentiability Implies Continuity

If a function $f:D\to \mathbb{R}$ is differentiable at $x_0\in D$, then $f$ is also continuous at $x_0$.

Proof of Differentiability Implies Continuity

Using the linear approximation form (Theorem 4.1.4): $f(x)=f(x_0)+\varphi (x)(x-x_0)$, where $\varphi$ is continuous at $x_0$.

Taking the limit as $x\to x_0$: ${\lim }_{x\to x_0}f(x)={\lim }_{x\to x_0}[f(x_0)+\varphi (x)(x-x_0)]=f(x_0)+{\lim }_{x\to x_0}\varphi (x)\cdot {\lim }_{x\to x_0}(x-x_0)=f(x_0)+\varphi (x_0)\cdot 0=f(x_0)$.

Thus ${\lim }_{x\to x_0}f(x)=f(x_0)$, which is the definition of continuity at $x_0$.

The converse is not true: continuity does not imply differentiability. A function can be continuous but have “corners” or “kinks” where the derivative does not exist (e.g., the absolute value function $f(x)=|x|$ is continuous at $x=0$, but not differentiable there).

Examples of Derivatives: Basic Building Blocks

Let’s calculate derivatives of some basic functions to get a feel for the definition.

Example 4.1.6 (1): Derivative of a Constant Function

Let $f(x)=1$ (the constant function). Then $f^{\prime }(x_0)=0$ for all $x_0\in \mathbb{R}$. The slope of a horizontal line is always zero.

Example 4.1.6 (2): Derivative of the Identity Function

Let $f(x)=x$ (the identity function). Then $f^{\prime }(x_0)=1$ for all $x_0\in \mathbb{R}$. The slope of the line $y=x$ is always 1.

Example 4.1.6 (3): Derivative of $x^2$

Let $f(x)=x^2$. Then $f^{\prime }(x_0)=2x_0$ for all $x_0\in \mathbb{R}$.

Proof that $(x^2{)}^{\prime }=2x$

Using the definition of the derivative: $f^{\prime }(x_0)={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}={\lim }_{x\to x_0}\frac{x^2-x_0^2}{x-x_0}={\lim }_{x\to x_0}\frac{(x-x_0)(x+x_0)}{x-x_0}={\lim }_{x\to x_0}(x+x_0)=2x_0$

Example 4.1.6 (4): Absolute Value Function is Not Differentiable at 0

Let $f(x)=|x|$. The derivative $f^{\prime }(0)$ does not exist. The function has a “corner” at $x=0$.

Example 4.1.6 (5): A Continuous Function Nowhere Differentiable

The function $g(x)=\min \{|x-m|:m\in \mathbb{Z}\}$ (a “sawtooth” function) is continuous everywhere but differentiable nowhere. Even more surprisingly, the function $f(x)={\sum }_{n=0}^{\infty }\frac{g(10^{n}x)}{10^n}$ (Van der Waerden function) is continuous everywhere but differentiable nowhere. This shows that there exist continuous functions that are “extremely non-smooth”.

These examples illustrate the definition of the derivative and highlight the relationship between differentiability and smoothness of functions. In the next sections, we will develop rules for computing derivatives more efficiently and explore the fundamental theorems that connect derivatives to the behavior of functions.

4.2 Central Theorems about the (First) Derivative: Making Differentiation Easier

Calculating derivatives directly from the definition can be tedious. Fortunately, there are rules that make differentiation much easier, especially for combinations of functions. These are the Differentiation Rules or “Rechenregeln” in the original script.

Theorem 4.1.9: Differentiation Rules

Let $D\subseteq \mathbb{R}$, $x_0\in D$ be a limit point of $D$, and let $f:D\to \mathbb{R}$ and $g:D\to \mathbb{R}$ be differentiable at $x_0$. Then:

1. Sum Rule: The sum $(f+g)$ is differentiable at $x_0$, and $(f+g{)}^{\prime }(x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$.
2. Product Rule (Leibniz Rule): The product $(f\cdot g)$ is differentiable at $x_0$, and $(f\cdot g{)}^{\prime }(x_0)=f^{\prime }(x_0)g(x_0)+f(x_0)g^{\prime }(x_0)$.
3. Quotient Rule: If $g(x_0)\ne 0$, then the quotient $(f/g)$ is differentiable at $x_0$, and ${\left(\frac{f}{g}\right)}^{\prime }(x_0)=\frac{f^{\prime }(x_0)g(x_0)-f(x_0)g^{\prime }(x_0)}{[g(x_0){]}^2}$

These rules allow us to differentiate sums, products, and quotients of functions by differentiating the individual functions and combining the results algebraically.

Proof of Differentiation Rules (using Linear Approximation - Theorem 4.1.3)

We use the linear approximation characterization of differentiability: $f(x)=f(x_0)+\varphi (x)(x-x_0)$ and $g(x)=g(x_0)+\psi (x)(x-x_0)$, where $\varphi$ and $\psi$ are continuous at $x_0$ and $\varphi (x_0)=f^{\prime }(x_0),\psi (x_0)=g^{\prime }(x_0)$.

1. Sum Rule: $(f+g)(x)=f(x)+g(x)=[f(x_0)+\varphi (x)(x-x_0)]+[g(x_0)+\psi (x)(x-x_0)]=[f(x_0)+g(x_0)]+[\varphi (x)+\psi (x)](x-x_0)$. Let $\xi (x)=\varphi (x)+\psi (x)$. Since $\varphi$ and $\psi$ are continuous at $x_0$, $\xi$ is also continuous at $x_0$ by the sum rule for continuous functions. And $\xi (x_0)=\varphi (x_0)+\psi (x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$. By Theorem 4.1.4, $(f+g)$ is differentiable at $x_0$ and $(f+g{)}^{\prime }(x_0)=\xi (x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$.

2. Product Rule: $(f\cdot g)(x)=f(x)g(x)=[f(x_0)+\varphi (x)(x-x_0)][g(x_0)+\psi (x)(x-x_0)]=f(x_0)g(x_0)+[f(x_0)\psi (x)+g(x_0)\varphi (x)+\varphi (x)\psi (x)(x-x_0)](x-x_0)$. Let $\zeta (x)=f(x_0)\psi (x)+g(x_0)\varphi (x)+\varphi (x)\psi (x)(x-x_0)$. Since $\varphi ,\psi$ are continuous at $x_0$, $\zeta$ is also continuous at $x_0$. And $\zeta (x_0)=f(x_0)\psi (x_0)+g(x_0)\varphi (x_0)+\varphi (x_0)\psi (x_0)(x_0-x_0)=f(x_0)g^{\prime }(x_0)+g(x_0)f^{\prime }(x_0)$. By Theorem 4.1.4, $(f\cdot g)$ is differentiable at $x_0$ and $(f\cdot g{)}^{\prime }(x_0)=\zeta (x_0)=f^{\prime }(x_0)g(x_0)+f(x_0)g^{\prime }(x_0)$.

3. Quotient Rule: (Proof involves similar but slightly more involved algebraic manipulation using the linear approximation form and showing the quotient fits the required form for differentiability).

Examples using Differentiation Rules:

Example 4.1.10 (1): Derivative of $x^n$

Using induction and the product rule, we can show that for $n\ge 1$, $(x^n{)}^{\prime }=n{x}^{n-1}$.

Example 4.1.10 (2): Derivative of Tangent Function

Using the quotient rule and the derivatives of sine and cosine, we can derive the derivative of the tangent function: $(\tan x{)}^{\prime }={\left(\frac{\sin x}{\cos x}\right)}^{\prime }=\frac{({\sin }^{\prime }x)\cos x-\sin x({\cos }^{\prime }x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$

4.3 The Chain Rule: Derivative of Composite Functions - Functions Inside Functions

The Chain Rule tells us how to differentiate a composite function, a function inside another function. If we have $g(f(x))$, the chain rule tells us how to find its derivative in terms of the derivatives of $f$ and $g$.

Theorem 4.1.11: Chain Rule

Let $D_1,D_2\subseteq \mathbb{R}$, $f:D_1\to D_2$, $g:D_2\to \mathbb{R}$. Let $x_0\in D_1$ and assume $f$ is differentiable at $x_0$, and $g$ is differentiable at $y_0=f(x_0)$. Then the composite function $(g\circ f):D_1\to \mathbb{R}$, defined by $(g\circ f)(x)=g(f(x))$, is differentiable at $x_0$, and its derivative is:

$(g\circ f{)}^{\prime }(x_0)=g^{\prime }(f(x_0))\cdot f^{\prime }(x_0)$

In words: The derivative of the outside function (evaluated at the inside function) times the derivative of the inside function.

Proof of Chain Rule (using Linear Approximation)

Let $y=f(x)$, $y_0=f(x_0)$. Since $f$ is differentiable at $x_0$, $f(x)-f(x_0)=\varphi (x)(x-x_0)$ with $\varphi (x_0)=f^{\prime }(x_0)$ and $\varphi$ continuous at $x_0$. Since $g$ is differentiable at $y_0=f(x_0)$, $g(y)-g(y_0)=\psi (y)(y-y_0)$ with $\psi (y_0)=g^{\prime }(y_0)$ and $\psi$ continuous at $y_0$.

Substitute $y=f(x)$ and $y_0=f(x_0)$ into the expression for $g$: $g(f(x))-g(f(x_0))=\psi (f(x))[f(x)-f(x_0)]=\psi (f(x))\varphi (x)(x-x_0)$.

Let $\xi (x)=\psi (f(x))\varphi (x)$. Since $f$ is continuous at $x_0$ (differentiability implies continuity), and $\psi$ and $\varphi$ are continuous at $f(x_0)$ and $x_0$ respectively, the composition $\psi (f(x))$ and the product $\psi (f(x))\varphi (x)=\xi (x)$ are continuous at $x_0$.

And $\xi (x_0)=\psi (f(x_0))\varphi (x_0)=g^{\prime }(f(x_0))f^{\prime }(x_0)$.

By Theorem 4.1.4, $(g\circ f)$ is differentiable at $x_0$ and $(g\circ f{)}^{\prime }(x_0)=\xi (x_0)=g^{\prime }(f(x_0))f^{\prime }(x_0)$.

Example: Derivative of $e^{\sin x}$.

Let $g(y)=e^y$ and $f(x)=\sin x$. Then $(g\circ f)(x)=e^{\sin x}$. Using the chain rule:

4.4 Inverse Functions and Their Derivatives

If a function is bijective (one-to-one and onto) and differentiable, and its inverse function is also continuous, we can find a formula for the derivative of the inverse function.

Corollary 4.1.12: Derivative of Inverse Function

Let $f:D\to E$ be a bijective function, $x_0\in D$ a limit point of $D$, and $y_0=f(x_0)$. Assume $f$ is differentiable at $x_0$ with $f^{\prime }(x_0)\ne 0$, and the inverse function $f^{-1}:E\to D$ is continuous at $y_0$. Then $f^{-1}$ is differentiable at $y_0$, and its derivative is:

$(f^{-1}{)}^{\prime }(y_0)=\frac{1}{f^{\prime }(x_0)}=\frac{1}{f^{\prime }(f^{-1}(y_0))}$

In words: The derivative of the inverse function at $y_0$ is the reciprocal of the derivative of the original function at $x_0=f^{-1}(y_0)$.

Proof of Derivative of Inverse Function

Let $y=f(x)$ and $y_0=f(x_0)$. Since $f$ is differentiable at $x_0$, $f(x)-f(x_0)=\varphi (x)(x-x_0)$ with $\varphi (x_0)=f^{\prime }(x_0)\ne 0$ and $\varphi$ continuous at $x_0$.

We want to find the derivative of $f^{-1}$ at $y_0$. Let $y\to y_0$. Since $f^{-1}$ is continuous at $y_0$, $x=f^{-1}(y)\to f^{-1}(y_0)=x_0$.

Rearranging the linear approximation formula: $x-x_0=\frac{1}{\varphi (x)}(f(x)-f(x_0))=\frac{1}{\varphi (f^{-1}(y))}(y-y_0)$.

So, $f^{-1}(y)-f^{-1}(y_0)=\frac{1}{\varphi (f^{-1}(y))}(y-y_0)$.

Let $\eta (y)=\frac{1}{\varphi (f^{-1}(y))}$. Since $f^{-1}$ and $\varphi$ are continuous at $y_0$ and $x_0$ respectively, and $\varphi (x_0)=f^{\prime }(x_0)\ne 0$, the function $\eta$ is continuous at $y_0$. And $\eta (y_0)=\frac{1}{\varphi (f^{-1}(y_0))}=\frac{1}{\varphi (x_0)}=\frac{1}{f^{\prime }(x_0)}$.

By Theorem 4.1.4, $f^{-1}$ is differentiable at $y_0$ and $(f^{-1}{)}^{\prime }(y_0)=\eta (y_0)=\frac{1}{f^{\prime }(x_0)}=\frac{1}{f^{\prime }(f^{-1}(y_0))}$.

Examples: Derivatives of Inverse Functions

• Derivative of Natural Logarithm: Since the exponential function $f(x)=e^x$ has derivative $f^{\prime }(x)=e^x$ and inverse function $f^{-1}(y)=\ln y$, we can use the inverse function theorem to find the derivative of $\ln y$:

  $$(\ln y{)}^{\prime }=\frac{1}{f^{\prime }(f^{-1}(y))}=\frac{1}{e^{\ln y}}=\frac{1}{y}$$

• Derivatives of Inverse Trigonometric Functions: We can use the inverse function theorem to find derivatives of arcsin, arccos, arctan, arccot, arcosh, arsinh, artanh, arccot, as shown in the script examples (Example 4.2.6 and 4.1.13).

These differentiation rules, including the Chain Rule and the Inverse Function Theorem, are the workhorses of differential calculus. They enable us to differentiate a vast range of functions and are essential tools for applications in mathematics, physics, engineering, and beyond.

4.5 Central Theorems About the First Derivative: Unlocking Function Behavior

Now we move to some of the most powerful theorems in differential calculus, theorems that reveal deep connections between the derivative of a function and its overall behavior. These are the Central Theorems about the (First) Derivative.

Rolle’s Theorem: Leveling Out

Rolle’s Theorem is a cornerstone result that sets the stage for the Mean Value Theorem. It deals with a function that starts and ends at the same height.

Theorem 4.2.3: Rolle's Theorem (1690)

Let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a)=f(b)$, then there exists at least one point $\xi \in (a,b)$ such that $f^{\prime }(\xi )=0$.

Imagine a smooth curve that starts and ends at the same y-value. Somewhere in between, the curve must “level out” – it must have a horizontal tangent line, meaning the derivative is zero.

Proof of Rolle's Theorem

Since $f$ is continuous on $[a,b]$, by the Min-Max Theorem (Extreme Value Theorem, Theorem 3.4.5), $f$ attains a maximum and a minimum value on $[a,b]$. Let $u,v\in [a,b]$ be points where the minimum and maximum are attained, respectively.

Case 1: $f$ is constant. If $f$ is constant on $[a,b]$, then $f^{\prime }(x)=0$ for all $x\in (a,b)$, so any $\xi \in (a,b)$ works.

Case 2: $f$ is not constant. Since $f(a)=f(b)$ and $f$ is not constant, either the maximum value or the minimum value (or both) must be attained at some point inside the interval $(a,b)$, not at the endpoints $a$ or $b$.

Suppose the maximum is attained at $\xi \in (a,b)$. Then $f$ has a local maximum at $\xi$. By Theorem 4.2.2(3), if $f$ has a local extremum (maximum or minimum) at an interior point $\xi$ where it’s differentiable, then $f^{\prime }(\xi )=0$. The same logic applies if the minimum is attained at an interior point.

Therefore, in either case, there exists a point $\xi \in (a,b)$ such that $f^{\prime }(\xi )=0$.

The Mean Value Theorem (MVT): The Average Slope

The Mean Value Theorem (MVT) is a generalization of Rolle’s Theorem. It says that for any smooth curve segment, there’s always a point where the tangent line is parallel to the secant line connecting the endpoints.

Theorem 4.2.4: Mean Value Theorem (Lagrange, 1797)

Let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists at least one point $\xi \in (a,b)$ such that:

$f(b)-f(a)=f^{\prime }(\xi )(b-a)$

Rearranging, we get:

The right side $\frac{f(b)-f(a)}{b-a}$ is the slope of the secant line connecting $(a,f(a))$ and $(b,f(b))$ – the average rate of change of $f$ over $[a,b]$. The MVT says that at some point $\xi$, the instantaneous rate of change $f^{\prime }(\xi )$ is equal to this average rate of change.

Proof of Mean Value Theorem (Using Rolle's Theorem)

Define a new function $h(x)$ which represents the vertical difference between the function $f(x)$ and the secant line connecting $(a,f(a))$ and $(b,f(b))$: $h(x)=f(x)-\left[\frac{f(b)-f(a)}{b-a}(x-a)+f(a)\right]$

1. Check that $h(a)=h(b)=0$: By construction, the secant line passes through $(a,f(a))$ and $(b,f(b))$, so the vertical difference is zero at the endpoints.
2. Apply Rolle’s Theorem to $h(x)$: Since $f$ and the linear function are continuous on $[a,b]$ and differentiable on $(a,b)$, so is $h$. And $h(a)=h(b)=0$. Thus, Rolle’s Theorem applies to $h$.
3. Conclude $h^{\prime }(\xi )=0$ for some $\xi \in (a,b)$: There exists $\xi \in (a,b)$ such that $h^{\prime }(\xi )=0$.
4. Compute $h^{\prime }(x)$: $h^{\prime }(x)=f^{\prime }(x)-\frac{f(b)-f(a)}{b-a}$.
5. Set $h^{\prime }(\xi )=0$ and rearrange: $0=f^{\prime }(\xi )-\frac{f(b)-f(a)}{b-a}$, which gives $f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}$, or $f(b)-f(a)=f^{\prime }(\xi )(b-a)$.

Consequences of the Mean Value Theorem: Linking Derivative to Function Behavior

The MVT is not just a theoretical curiosity; it’s a powerful tool with many important consequences. It allows us to relate the derivative of a function (local information) to its global behavior (monotonicity, boundedness, etc.).

Corollary 4.2.5: Consequences of the Mean Value Theorem

Let $f,g:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$.

1. Zero Derivative Implies Constant Function: If $f^{\prime }(\xi )=0$ for all $\xi \in (a,b)$, then $f$ is constant on $[a,b]$.
2. Equal Derivatives Imply Constant Difference: If $f^{\prime }(\xi )=g^{\prime }(\xi )$ for all $\xi \in (a,b)$, then there exists a constant $C\in \mathbb{R}$ such that $f(x)=g(x)+C$ for all $x\in [a,b]$.
3. Non-negative Derivative Implies Increasing Function: If $f^{\prime }(\xi )\ge 0$ for all $\xi \in (a,b)$, then $f$ is monotonically increasing on $[a,b]$.
4. Positive Derivative Implies Strictly Increasing Function: If $f^{\prime }(\xi )>0$ for all $\xi \in (a,b)$, then $f$ is strictly monotonically increasing on $[a,b]$.
5. Non-positive Derivative Implies Decreasing Function: If $f^{\prime }(\xi )\le 0$ for all $\xi \in (a,b)$, then $f$ is monotonically decreasing on $[a,b]$.
6. Negative Derivative Implies Strictly Decreasing Function: If $f^{\prime }(\xi )<0$ for all $\xi \in (a,b)$, then $f$ is strictly monotonically decreasing on $[a,b]$.
7. Bounded Derivative Implies Lipschitz Continuity: If there exists $M\ge 0$ such that $|f^{\prime }(\xi )|\le M$ for all $\xi \in (a,b)$, then for all $x_1,x_2\in [a,b]$, $|f(x_1)-f(x_2)|\le M|x_1-x_2|$. (Lipschitz continuity implies uniform continuity).

These corollaries are incredibly useful. They allow us to determine if a function is constant, increasing, or decreasing simply by examining the sign of its derivative. They also provide bounds on how much a function can change based on the boundedness of its derivative.

4.6 Higher Derivatives: Rate of Change of Rate of Change

We can differentiate a function more than once. The second derivative is the derivative of the first derivative, the third derivative is the derivative of the second derivative, and so on. These higher derivatives give us information about the rate of change of the rate of change, and so on, capturing more subtle aspects of a function’s behavior, like concavity and inflection points (which we’ll explore later).

Definition 4.3.1: Higher Derivatives

1. For $n\ge 2$, a function $f:D\to \mathbb{R}$ is $n$-times differentiable on $D$ if its $(n-1)$-th derivative $f^{(n-1)}$ is differentiable on $D$. The $n$-th derivative is denoted by $f^{(n)}=(f^{(n-1)}{)}^{\prime }$. We also write $f^{(1)}=f^{\prime }$ and $f^{(0)}=f$.
2. A function $f$ is $n$-times continuously differentiable on $D$ if it is $n$-times differentiable on $D$ and its $n$-th derivative $f^{(n)}$ is continuous on $D$.
3. A function $f$ is smooth (or infinitely differentiable) on $D$ if it is $n$-times differentiable for all $n\ge 1$.

Notation:

• $f^{\prime }(x),f^{\prime \prime }(x),f^{\prime \prime \prime }(x),...$ for the first, second, third, … derivatives.
• $f^{(1)}(x),f^{(2)}(x),f^{(3)}(x),...$ for the first, second, third, … derivatives.
• $f^{(n)}(x)$ for the $n$-th derivative.

Rules for Higher Derivatives:

We can extend the differentiation rules (Sum Rule, Product Rule, Chain Rule, Quotient Rule) to higher derivatives. For example, Leibniz’s Rule generalizes the Product Rule to higher derivatives:

Theorem 4.3.3: Leibniz Rule for Higher Derivatives of a Product

Let $f,g:D\to \mathbb{R}$ be $n$-times differentiable on $D$. Then their product $(f\cdot g)$ is also $n$-times differentiable on $D$, and its $n$-th derivative is given by:

$(f\cdot g{)}^{(n)}={\sum }_{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})f^{(k)}{g}^{(n-k)}$

This formula looks similar to the Binomial Theorem and is proved using induction.

Examples of Higher Derivatives:

• Exponential Function: The exponential function $f(x)=e^x$ is infinitely differentiable, and all its derivatives are equal to itself: $f^{(n)}(x)=e^x$ for all $n\ge 1$.
• Polynomials: Polynomials are also infinitely differentiable. However, their higher derivatives eventually become zero. For a polynomial of degree $n$, its $(n+1)$-th derivative and all higher derivatives are zero.
• Natural Logarithm: The natural logarithm $\ln (x)$ (for $x>0$) is infinitely differentiable, with derivatives given by a pattern: $(\ln {)}^{(n)}(x)=(-1{)}^{n-1}(n-1)!x^{-n}$ for $n\ge 1$.

4.7 Taylor Approximation: Polynomials That Mimic Functions

Taylor approximation is a powerful technique that uses polynomials to approximate functions. The idea is to find a polynomial that “mimics” the behavior of a function near a particular point, matching its value and its derivatives at that point.

Theorem 4.4.5: Taylor's Theorem (with Remainder in Lagrange Form)

Let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and $(n+1)$-times differentiable on $(a,b)$. For any $x\in [a,b]$ and a fixed point $a\in [a,b]$, there exists a point $\xi$ between $a$ and $x$ such that:

$f(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k+R_n(x)$

where the remainder term $R_n(x)$ is given by:

$R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-a{)}^{n+1}$

The polynomial $T_n(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k$ is called the Taylor polynomial of degree $n$ for $f$ centered at $a$. It’s the best polynomial approximation of $f$ near $x=a$ in the sense that it matches the function’s value and first $n$ derivatives at $a$. The remainder term $R_n(x)$ quantifies the error in this approximation.

Taylor Series:

If a function is infinitely differentiable, we can consider the Taylor series:

This is an infinite series that is constructed from the function’s derivatives at $a$. For many important functions (like $e^x,\sin x,\cos x,\ln (1+x)$), the Taylor series actually converges to the function itself within its radius of convergence. This means we can represent these functions exactly as infinite polynomials (power series).

Taylor approximation and Taylor series are fundamental tools in analysis, approximation theory, numerical analysis, and many other areas. They allow us to approximate complicated functions with simpler polynomials, analyze function behavior, and solve problems that would be intractable otherwise.

This concludes Chapter 4. We’ve explored the derivative, its rules, its connection to function behavior through the Mean Value Theorem, higher derivatives, and the powerful tool of Taylor approximation. These concepts are essential for understanding and applying calculus in a wide range of contexts.

Previous Chapter: Chapter 3 - Continuous Functions, Smoothness and Limits, Continuity, Functions Without Jumps, Combining Continuous Functions, Fundamental Theorems of Continuity, Exponential and Trigonometric Functions

Next Chapter: Chapter 5 - The Riemann Integral, Measuring Areas Under Curves, Properties and Classes of Integrable Functions, Properties of Integrals, Linearity, Monotonicity, and the Mean Value Theorem for Integrals, Fundamental Theorem and Applications