Calculus Bible Chapter 2: Derivatives

This post summarizes Chapter 2 of James Stewart's Calculus.

2.1 Derivatives and Rates of Change#

The Tangent Problem (p.80)#

The slope $m$ of a tangent line is expressed as the limit of the slope of a secant line.

The tangent line to the curve $y=f(x)$ at the point $P(a, f(a))$ is the line through $P$ with slope (if the limit exists):

$m=\lim_{x \to a} \frac{f(x) - f(a)}{x-a}$

If we let $h = x - a$, then $x = a + h$, and the slope of the secant line $PQ$ can be written as:

$m_{PQ}=\frac{f(a + h)-f(a)}{h}$

The Velocity Problem (p.81)#

The function $f$ describing the motion of an object is called its position function. Over the time interval from $t=a$ to $t=a+h$, the change in position is $f(a+h)-f(a)$.

The average velocity over this interval is:

$\text{Average velocity} = \frac{\text{displacement}}{\text{time}} = \frac{f(a+h) - f(a)}{h}$

Now suppose we compute the average velocity over shorter and shorter time intervals $[a, a+h]$ — that is, as $h$ approaches $0$.

The instantaneous velocity $v(a)$ at time $t=a$ is defined as the limit of average velocities: (p.82)

$v(a)=\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

The Derivative at a Point (p.83)#

The derivative$^{derivative}$ of a function $f$ at a number $a$, denoted $f'(a)$, is defined by the following limit (if it exists):

$f'(a)=\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Rates of Change#

Let $y$ be a quantity that depends on another quantity $x$. Then $y$ is a function of $x$, written $y=f(x)$. If $x$ changes from $x_1$ to $x_2$, the change in $x$ (the increment of $x$) is:

$\Delta x = x_2 - x_1$

The corresponding change in $y$ is:

$\Delta y = f(x_2) - f(x_1)$

The following difference quotient is called the average rate of change of $y$ with respect to $x$ over the interval $[x_1, x_2]$: (p.84)

$\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

The limit of this average rate of change is called the instantaneous rate of change of $y$ with respect to $x$ at $x = x_1$:

$\text{Instantaneous rate of change} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{x_2 \to x_1} \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

2.2 The Derivative as a Function#

If we replace the fixed number $a$ with a variable $x$ in the derivative definition, we obtain: (p.88)

$f'(x)=\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

This function is called the derivative of $f$.

Alternative notations for the derivative (p.92)

$f'(x)=y'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f(x)=Df(x)=D_xf(x)$

The symbols $D$ and $\frac{d}{dx}$ are called differentiation operators — they denote the operation of differentiation, the process of computing a derivative. The notation $\frac{dy}{dx}$ was introduced by Leibniz.

Differentiable Functions (p.92)

A function $f$ is differentiable at $a$ if $f'(a)$ exists. $f$ is differentiable on an open interval $(a, b)$ [or $(a, \infty),\ (-\infty, a),\ (-\infty, \infty)$] if it is differentiable at every number in the interval.

The relationship between continuity and differentiability is: (p.93)

If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

When a function is not differentiable (p.94)

Three cases where a function fails to be differentiable:

• A corner (sharp bend)
• A discontinuity
• A vertical tangent (perfectly vertical slope)

Higher Derivatives (p.94)

If $f$ is a differentiable function, then its derivative $f'$ is also a function, so we can compute $(f')'=f''$. This is called the second derivative.

In Leibniz notation, the second derivative of $y=f(x)$ is written:

$\frac{d}{dx}\!\left(\frac{dy}{dx}\right)=\frac{d^2y}{dx^2}$

c.f. The instantaneous rate of change of velocity with respect to time is called the acceleration $a(t)$ of the object. The acceleration function is the derivative of the velocity function, and the second derivative of the position function:

$a(t)=v'(t)=s''(t)$

Third Derivative (p.97)

The third derivative can be interpreted as the slope of the curve $y=f''(x)$, or as the rate of change of $f''(x)$. For $y=f(x)$, the notation for the third derivative is:

$y'''=f'''(x)=\frac{d}{dx}\!\left(\frac{d^2y}{dx^2}\right)=\frac{d^3y}{dx^3}$

2.3 Basic Differentiation Rules#

Derivative of a Constant Function (p.100)

$\frac{d}{dx}(c)=0$

The Power Rule — If $n$ is a positive integer, then: (p.100)

$\frac{d}{dx}(x^n)=nx^{n-1}$

The Constant Multiple Rule — If $c$ is a constant and $f$ is a differentiable function, then: (p.104)

$\frac{d}{dx}[c \cdot f(x)]=c\frac{d}{dx}f(x)$

The Sum Rule — If $f$ and $g$ are differentiable functions, then: (p.104)

$\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$

The Difference Rule — If $f$ and $g$ are differentiable functions, then: (p.104)

$\frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}f(x)-\frac{d}{dx}g(x)$

Derivatives of Sine and Cosine (p.106)

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

2.4 The Product and Quotient Rules#

The Product Rule — If $f$ and $g$ are differentiable functions, then: (p.112)

$\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

The Quotient Rule — If $f$ and $g$ are differentiable functions, then: (p.112)

$\frac{d}{dx}\!\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\dfrac{d}{dx}[f(x)]-f(x)\dfrac{d}{dx}[g(x)]}{[g(x)]^2}$

Derivatives of Trigonometric Functions (p.116)

$\frac{d}{dx}(\sin x)=\cos x \qquad \frac{d}{dx}(\cos x)=-\sin x \qquad \frac{d}{dx}(\tan x)=\sec^2 x$

$\frac{d}{dx}(\csc x)=-\csc x \cot x \qquad \frac{d}{dx}(\sec x)=\sec x \tan x \qquad \frac{d}{dx}(\cot x)=-\csc^2 x$

2.5 The Chain Rule#

The differentiation rules covered so far cannot handle composite functions. (p.119)

Example: $F(x)=\sqrt{x^2+1}$

If we let $y=f(u)=\sqrt{u}$ and $u=g(x)=x^2+1$, then $y=F(x)=f(g(x))$, i.e., $F=f\circ g$.

The derivative of the composite function $f\circ g$ is the product of the derivatives of $f$ and $g$. This is the Chain Rule.

Intuition: If $\frac{du}{dx}$ is the rate of change of $u$ with respect to $x$, $\frac{dy}{du}$ is the rate of change of $y$ with respect to $u$, and $\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$, then if $u$ changes twice as fast as $x$ and $y$ changes three times as fast as $u$, then $y$ changes six times as fast as $x$.

Leibniz notation:

$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Prime notation:

$F'(x) = f'(g(x))\cdot g'(x)$

The Power Rule Combined with the Chain Rule (p.123)

$\frac{d}{dx}(u^n)=nu^{n-1}\frac{du}{dx} \qquad \text{or} \qquad \frac{d}{dx}[g(x)]^n =n[g(x)]^{n-1}\cdot g'(x)$

2.6 Implicit Differentiation#

• Explicit function$^{explicit\ function}$: a function where one variable is defined directly in terms of another (e.g., $y=\sqrt{x^3+1}$) (p.128)
• Implicit function$^{implicit\ function}$: a function defined by a relation between multiple variables (e.g., $x^2+y^2=25$)

To differentiate an implicit function, we use implicit differentiation: differentiate both sides with respect to $x$, then solve for $y'$.

Example: Differentiating $x^2+y^2=25$ using implicit differentiation (p.129)

1. Differentiate both sides.

$\frac{d}{dx}(x^2 + y^2)=\frac{d}{dx}(25)$

$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2)=0$

2. Since $y^2$ is not directly a function of $x$, apply the chain rule by treating $y$ as a function of $x$.

$\frac{d}{dx}(y^2)=\frac{d}{dy}(y^2)\frac{dy}{dx}=2y\frac{dy}{dx}$

$2x+2y\frac{dy}{dx}=0$

3. Solving for $\dfrac{dy}{dx}$ gives:

$\frac{dy}{dx}=-\frac{x}{y}$

2.7 Related Rates#

Related Rates Problems (p.134): Given two quantities whose rates of change are related, use the rate of change of the more easily measured quantity to find the rate of change of the other.

Example: Air is being pumped into a spherical balloon so that its volume is increasing at a rate of $100\ \text{cm}^3/\text{s}$. How fast is the radius increasing when the diameter is $50\ \text{cm}$?

1. Identify the given and unknown information
   • Given: the rate of increase of volume is $100\ \text{cm}^3/\text{s}$
   • Unknown: the rate of increase of the radius when the diameter is $50\ \text{cm}$
2. Let $V$ be the volume of the balloon and $r$ the radius. Both are functions of time $t$.
   • Given: $\dfrac{dV}{dt}=100\ \text{cm}^3/\text{s}$
   • Unknown: $\dfrac{dr}{dt}$ when $r=25\ \text{cm}$
3. To relate $\dfrac{dV}{dt}$ and $\dfrac{dr}{dt}$, use the formula for the volume of a sphere to connect $V$ and $r$.

$V=\frac{4}{3}\pi r^3$

4. Differentiate both sides with respect to $t$. Use the chain rule on the right side.

$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$

5. Solve for the unknown quantity.

$\frac{dr}{dt}=\frac{1}{4\pi r^2}\frac{dV}{dt}$

6. Substitute $r=25\ \text{cm}$ and $\dfrac{dV}{dt}=100$.

$\frac{dr}{dt}=\frac{1}{4\pi (25)^2}\cdot 100=\frac{1}{25\pi}$

2.8 Linear Approximations and Differentials#

When $x$ is near $a$, we use the tangent line to $f$ at the point $(a, f(a))$ as an approximation to the curve $y=f(x)$. (p.140)

Equation of the tangent line:

$y=f(a)+f'(a)(x-a)$

The following approximation is called the linear approximation$^{linear\ approximation}$ or tangent line approximation$^{tangent\ line\ approximation}$ of $f$ at $a$:

$f(x)\approx f(a) + f'(a)(x-a)$

In other words, a single tangent line lets us approximate the value of the function at points near the point of tangency.

This is also widely applied in physics. (p.143)

Summary#

• Derivative at a point: $f'(a)=\lim\limits_{h \to 0} \dfrac{f(a+h)-f(a)}{h}$ — the slope of the tangent line and the instantaneous rate of change
• Derivative as a function: $f'(x)$, obtained by replacing the fixed number $a$ with $x$
• Basic differentiation rules: constant, power, constant multiple, sum, difference, trigonometric functions
• Product Rule / Quotient Rule: differentiating products or quotients of two functions
• Chain Rule: differentiating composite functions — $F'(x)=f'(g(x))\cdot g'(x)$
• Implicit differentiation: differentiate both sides with respect to $x$, then solve for $y'$
• Related rates: use the chain rule to connect rates of change of different quantities
• Linear approximation: approximate function values using the tangent line