Lesson 4: Basic Differentiation Rules
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Apply the constant rule, power rule, and constant multiple rule
- Differentiate sums and differences of functions
- Find derivatives of polynomials quickly
- Differentiate ex and ln x
- Solve problems involving rates of change using these rules
The Constant Rule
Constant Rule: If f(x) = c (a constant), then f'(x) = 0.
The graph of a constant is a horizontal line, so its slope is zero everywhere.
The Power Rule
Power Rule: If f(x) = xn for any real number n, then f'(x) = nxn−1.
This works for positive integers, negative integers, and even fractions (rational exponents).
Example 1: Power Rule Applications
(a) d/dx [x5] = 5x4
(b) d/dx [x−3] = −3x−4 = −3/x4
(c) d/dx [√x] = d/dx [x1/2] = (1/2)x−1/2 = 1/(2√x)
(d) d/dx [x] = 1x0 = 1
Constant Multiple Rule
Constant Multiple Rule: d/dx [c · f(x)] = c · f'(x).
You can "factor out" constants when differentiating.
Example 2: Constant Multiples
(a) d/dx [7x4] = 7 · 4x3 = 28x3
(b) d/dx [−3x2] = −3 · 2x = −6x
(c) d/dx [(4/3)x6] = (4/3)(6x5) = 8x5
Sum and Difference Rules
Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
Difference Rule: d/dx [f(x) − g(x)] = f'(x) − g'(x)
Differentiate term by term.
Example 3: Differentiating a Polynomial
Find d/dx [3x4 − 2x3 + 7x − 5].
Solution: Differentiate each term:
= 3(4x3) − 2(3x2) + 7(1) − 0 = 12x3 − 6x2 + 7
Example 4: Rewriting Before Differentiating
Find f'(x) for f(x) = (3x² + 2)/x.
Step 1: Rewrite: f(x) = 3x + 2x−1
Step 2: f'(x) = 3 + 2(−1)x−2 = 3 − 2/x²
Derivatives of Exponential and Logarithmic Functions
Exponential: d/dx [ex] = ex (the only function that is its own derivative!)
Natural Logarithm: d/dx [ln x] = 1/x for x > 0
Example 5: Mixed Functions
Find d/dx [5ex − 3 ln x + x4].
= 5ex − 3/x + 4x3
Finding Tangent Lines with Rules
Example 6: Tangent Line
Find the tangent line to f(x) = x3 − 4x + 2 at x = 1.
Step 1: f(1) = 1 − 4 + 2 = −1. Point: (1, −1).
Step 2: f'(x) = 3x² − 4. f'(1) = 3 − 4 = −1.
Step 3: y − (−1) = −1(x − 1), so y = −x.
Higher-Order Derivatives
The derivative of the derivative is called the second derivative: f''(x) or d²y/dx². You can continue: f'''(x), f(4)(x), etc.
Example 7: Higher Derivatives
f(x) = x5 − 3x3 + x
f'(x) = 5x4 − 9x² + 1
f''(x) = 20x3 − 18x
f'''(x) = 60x² − 18
Key Takeaways
- Power Rule: d/dx [xn] = nxn−1 — the most-used rule in calculus.
- Constant, sum, difference, constant multiple rules let you differentiate any polynomial term by term.
- d/dx [ex] = ex and d/dx [ln x] = 1/x.
- When a function is a fraction, rewrite using negative exponents before differentiating (for now — the quotient rule comes in Module 3).
- Higher-order derivatives describe rates of rates (acceleration is the second derivative of position).
Check Your Understanding
1. Find f'(x) for f(x) = 4x3 − 2x + 9.
2. Find g'(x) for g(x) = 5/x² + 3√x.
3. Find the second derivative of h(x) = ex + x4.
4. Find the tangent line to y = 2ex at x = 0.