Unit 2
Differentiation: Definition and Fundamental Properties
Notes
The derivative as a limit
The derivative is the instantaneous rate of change, the slope of the tangent line:
- The average rate of change $\displaystyle \frac{f(b) - f(a)}{b - a}$ is a secant slope. As $b \to a$, secants approach the tangent.
- Notation: $f'(x)$, $\displaystyle \frac{dy}{dx}$, $y'$, $\displaystyle \frac{d}{dx}f(x)$.
- Tangent line at $x = a$: $y = f(a) + f'(a)(x - a)$.
Differentiability and continuity
- Differentiable at $a$ implies continuous at $a$. The reverse is false: $|x|$ is continuous at 0 but has a corner there.
- A function is not differentiable at a corner, a cusp, a vertical tangent, or any discontinuity.
Basic rules
| Rule | Derivative |
|---|---|
| Constant | $\displaystyle \frac{d}{dx}c = 0$ |
| Power | $\displaystyle \frac{d}{dx}x^n = nx^{n-1}$ |
| Constant multiple | $\displaystyle \frac{d}{dx}[cf] = cf'$ |
| Sum / difference | $\displaystyle \frac{d}{dx}[f \pm g] = f' \pm g'$ |
| Product | $\displaystyle \frac{d}{dx}[fg] = f'g + fg'$ |
| Quotient | $\displaystyle \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$ |
- $\displaystyle \frac{d}{dx}\sin x = \cos x$, $\displaystyle \frac{d}{dx}\cos x = -\sin x$, $\displaystyle \frac{d}{dx}e^x = e^x$, $\displaystyle \frac{d}{dx}\ln x = \frac{1}{x}$.
- $\displaystyle \frac{d}{dx}\tan x = \sec^2 x$, $\displaystyle \frac{d}{dx}\cot x = -\csc^2 x$, $\displaystyle \frac{d}{dx}\sec x = \sec x\tan x$, $\displaystyle \frac{d}{dx}\csc x = -\csc x\cot x$.
Topics in this unit
Each topic has its own short notes and practice.
- 2.1Defining Average and Instantaneous Rates of Change at a Point
- 2.2Defining the Derivative of a Function and Using Derivative Notation
- 2.3Estimating Derivatives of a Function at a Point
- 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5The Power Rule
- 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7Derivatives of cos x, sin x, e^x, and ln x
- 2.8The Product Rule
- 2.9The Quotient Rule
- 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Practice
$f(3 + h) - f(3) = (3 + h)^2 - 9$, divided by $h$.
$\displaystyle \frac{2}{x} = 2x^{-1}$ has derivative $-2x^{-2}$, so subtracting it gives $\displaystyle +\frac{2}{x^2}$.
$y' = 3x^2 - 2 = 3(4) - 2 = 10$.
The graph has a corner at $x = 1$: the slopes from the left ($-1$) and right ($1$) differ.
Product rule: $(x^2)'\sin x + x^2(\sin x)'$.
Quotient rule: $\displaystyle \frac{1(x + 1) - x(1)}{(x + 1)^2} = \frac{1}{(x+1)^2}$.
It's the definition of the derivative of $e^x$ at $x = 2$, which is $e^2$.
$y(4) = 2$ and $\displaystyle y' = \frac{1}{2\sqrt x} = \frac{1}{4}$ at $x = 4$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
The table gives the temperature $T(t)$, in °C, of a pan $t$ minutes after it is placed on a stove: $T(0) = 20$, $T(2) = 44$, $T(5) = 71$, $T(9) = 95$.
(a) Estimate $T'(3.5)$ using the table. Show the work and give units.
Show the answer and scoring guide
$\displaystyle T'(3.5) \approx \frac{T(5) - T(2)}{5 - 2} = \frac{71 - 44}{3} = 9$ °C per minute.
Check off the points you earned:
(b) Find the average rate of change of $T$ over $[0, 9]$ and interpret it.
Show the answer and scoring guide
$\displaystyle \frac{95 - 20}{9} = \frac{75}{9} \approx 8.33$. On average, the pan's temperature rose about 8.33 °C per minute during the first 9 minutes.
Check off the points you earned:
(c) A model for the temperature is $M(t) = 100 - 80e^{-0.15t}$. Find $M'(t)$.
Show the answer and scoring guide
$M'(t) = -80 \cdot (-0.15)e^{-0.15t} = 12e^{-0.15t}$.
Check off the points you earned: