This study guide covers the fundamental concepts of derivatives, differentiation, and their applications as outlined in Chapter 2. Understanding derivatives is essential for studying rates of change, optimization, and many real-world applications.
The derivative measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line at a point.
Derivative = Instantaneous rate of change = Slope of tangent line
The concept of derivatives was independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.
Newton: ẏ (y-dot)
Leibniz: dy/dx
Lagrange: f'(x)
Cauchy: Df(x)
• Velocity = derivative of position
• Acceleration = derivative of velocity
• Marginal cost in economics
• Rate of chemical reactions
Derivatives help us understand how things change in the real world.
Key Definition: The derivative of f at x, denoted f'(x), is defined as:
f'(x) = limh→0 [f(x+h) – f(x)]/h
This is called differentiation from first principles or by definition.
The average rate of change of f(x) over the interval [x₁, x₂] is:
This represents the slope of the secant line through points (x₁, f(x₁)) and (x₂, f(x₂)).
Average rate = (Change in y) / (Change in x) = Δy/Δx
The instantaneous rate of change at x = a is:
This represents the slope of the tangent line at x = a.
Example: If f(x) = x², find the average rate of change over [1, 3] and instantaneous rate at x = 2.
Solution:
Average rate = [f(3)-f(1)]/(3-1) = (9-1)/2 = 4
Instantaneous rate: f'(x) = 2x, so f'(2) = 4
The derivative f'(a) gives the slope of the tangent line to the curve y = f(x) at x = a.
Tangent line: y – f(a) = f'(a)(x – a)
Secant line connects two points on the curve. As the points get closer, the secant approaches the tangent line.
Think of zooming in on a curve until it looks like a straight line – that’s the tangent!
The derivative from first principles:
1. Find f(x+h)
2. Compute f(x+h) – f(x)
3. Divide by h
4. Take limit as h→0
If f is differentiable at x = a, then f is continuous at x = a.
But the converse is not true! A function can be continuous but not differentiable.
A function may not be differentiable at:
f(x) = |x| is not differentiable at x = 0 (corner)
| Function | Derivative | Domain Condition |
|---|---|---|
| f(x) = c (constant) | f'(x) = 0 | All real numbers |
| f(x) = xⁿ | f'(x) = nxⁿ⁻¹ | All real numbers (for integer n) |
| f(x) = eˣ | f'(x) = eˣ | All real numbers |
| f(x) = aˣ | f'(x) = aˣ ln a | All real numbers (a>0, a≠1) |
| f(x) = ln x | f'(x) = 1/x | x > 0 |
| f(x) = logₐx | f'(x) = 1/(x ln a) | x > 0, a>0, a≠1 |
“Bring down the power, reduce power by 1”
Constants come out front when differentiating.
Derivative of sum = sum of derivatives.
“Derivative of first times second, plus first times derivative of second”
“Low d-high minus high d-low, over low squared”
(Low = denominator, High = numerator)
Or in Leibniz notation: dy/dx = dy/du · du/dx
“Derivative of outside (keeping inside same) times derivative of inside”
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| cot x | -csc² x |
| sec x | sec x tan x |
| csc x | -csc x cot x |
The derivatives of “co-” functions (cos, cot, csc) start with a negative sign.
d/dx (sin⁻¹x) = 1/√(1-x²)
d/dx (cos⁻¹x) = -1/√(1-x²)
d/dx (tan⁻¹x) = 1/(1+x²)
For sin⁻¹x and cos⁻¹x: denominator is √(1-x²)
For tan⁻¹x and cot⁻¹x: denominator is 1+x²
cos⁻¹x, cot⁻¹x, csc⁻¹x have negative derivatives
Useful for functions of the form f(x) = u(x)^(v(x)) or products of many functions.
1. Take ln of both sides
2. Use log properties to simplify
3. Differentiate implicitly
4. Solve for dy/dx
When y is not explicitly expressed as a function of x:
1. Differentiate both sides with respect to x
2. Remember: d/dx (y) = dy/dx
3. Solve for dy/dx
If x = f(t) and y = g(t), then:
f is increasing on interval I if f'(x) > 0 for all x in I.
Graph rises as x increases.
f is decreasing on interval I if f'(x) < 0 for all x in I.
Graph falls as x increases.
A critical point occurs where f'(x) = 0 or f'(x) does not exist.
Critical points are candidates for local maxima or minima.
• Stationary point: f'(x) = 0
• Corner/cusp: f'(x) doesn’t exist
• Vertical tangent: f'(x) = ∞
To classify critical points:
1. Find critical points (where f'(x)=0 or DNE)
2. Check sign of f'(x) on either side:
• f’ changes + to – → local maximum
• f’ changes – to + → local minimum
• f’ doesn’t change sign → neither
At a point where f'(c) = 0:
• If f”(c) > 0 → local minimum
• If f”(c) < 0 → local maximum
• If f”(c) = 0 → test inconclusive (use first derivative test)
f”(x) > 0 → graph curves upward
Tangent lines lie below the graph
f”(x) < 0 → graph curves downward
Tangent lines lie above the graph
Where concavity changes (f”(x) changes sign)
f”(x) = 0 or f”(x) doesn’t exist
Finding maximum or minimum values of a function.
1. Understand the problem
2. Draw diagram if helpful
3. Write equation for quantity to optimize
4. Reduce to single variable if needed
5. Find critical points
6. Use first or second derivative test
7. Check endpoints if domain is closed interval
Finding rate of change of one quantity knowing rate of change of related quantity.
1. Identify variables and rates
2. Write equation relating variables
3. Differentiate with respect to time (t)
4. Substitute known values
5. Solve for unknown rate
Successive differentiation:
Applications:
f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …
Expansion about x = 0
f(x) = f(a) + f'(a)(x-a) + f”(a)(x-a)²/2! + …
Expansion about x = a
Test your knowledge with these 50 multiple choice questions. Select your answer and check the explanation.
• Memorize basic derivative formulas
• Practice product, quotient, and chain rules
• Work with trigonometric and exponential functions
Create flashcards for derivative formulas and practice daily.
• Connect derivatives to real-world problems
• Practice optimization and related rates
• Interpret graphs using derivatives
For each application, ask: “What does the derivative represent here?”
• Work implicit differentiation problems
• Practice logarithmic differentiation
• Solve parametric and higher-order derivatives
Start with simple problems and gradually increase difficulty.
1. Read each question carefully
2. Identify which rule(s) to apply
3. Check your work by differentiating again
4. Verify domain restrictions
5. For applications, include units in final answer
• Limit definition of derivative
• Power rule and basic derivatives
• Sum/difference and constant multiple rules
• Product and quotient rules
• Chain rule and composite functions
• Trigonometric derivatives
• Implicit differentiation
• Logarithmic differentiation
• Parametric differentiation
• Increasing/decreasing functions
• Maxima and minima problems
• Related rates and optimization
“First d-second + second d-first”
Or: “Left d-right + right d-left”
“Low d-high minus high d-low, over low squared”
Or: “Bottom d-top minus top d-bottom, over bottom squared”
“Derivative of outside (inside unchanged) times derivative of inside”
Or: “Work from outside in”
Final Advice: Derivatives are the foundation of calculus. Master these concepts thoroughly, as they will be used extensively in integration, differential equations, and advanced mathematics. Practice regularly, understand the “why” behind each rule, and don’t hesitate to revisit fundamentals when needed.