The Power Rule
Learn the shortcut for differentiating powers of x, and apply the constant multiple rule and sum/difference rule to differentiate any polynomial.
The Power Rule
The power rule is the most frequently used differentiation rule. It provides a shortcut that avoids the lengthy first-principles process for power functions.
If f(x) = xn, then
f′(x) = nxn−1
"Bring the power down, then reduce the power by 1"
Pattern Recognition
| f(x) | f′(x) | Explanation |
|---|---|---|
| x² | 2x | 2x2−1 = 2x1 = 2x |
| x³ | 3x² | 3x3−1 = 3x² |
| x5 | 5x4 | 5x5−1 = 5x4 |
| x | 1 | x = x1, so 1·x0 = 1 |
| x−1 = 1/x | −x−2 | −1·x−2 = −1/x² |
| x1/2 = √x | 1/2x−1/2 | 1/2x−1/2 = 1/2√x |
Constant Rule and Constant Multiple Rule
Constant Rule
If f(x) = c, then f′(x) = 0
The derivative of any constant is zero. A constant function has no rate of change.
Constant Multiple Rule
d/dx[c · f(x)] = c · f′(x)
Constants can be factored out of the derivative. Multiply by the constant after differentiating.
Examples:
- d/dx [5x³] = 5 × 3x² = 15x²
- d/dx [−2x4] = −2 × 4x³ = −8x³
- d/dx [1/3x6] = 1/3 × 6x5 = 2x5
Sum and Difference Rule
To differentiate a sum (or difference) of functions, simply differentiate each term separately:
d/dx[f(x) ± g(x)] = f′(x) ± g′(x)
This means any polynomial can be differentiated term by term. For example:
Differentiate: y = 4x³ − 2x² + 7x − 5
dy/dx = 4(3x²) − 2(2x) + 7(1) − 0
dy/dx = 12x² − 4x + 7
Key Vocabulary
Power Rule
If f(x) = xn, then f′(x) = nxn−1. Works for any real number n.
Polynomial
A function with terms of the form axn, where n is a non-negative integer.
Differentiate
The process of finding the derivative of a function.
Gradient Function
Another name for the derivative — a function that gives the slope at each point.
Worked Examples
Differentiate f(x) = 6x4 − 3x² + 8.
Step 1: Apply the power rule to each term: d/dx[6x4] = 6 × 4x³ = 24x³.
Step 2: d/dx[−3x²] = −3 × 2x = −6x.
Step 3: d/dx[8] = 0 (constant rule).
Answer: f′(x) = 24x³ − 6x.
Find dy/dx if y = 3/x² + √x.
Step 1: Rewrite: y = 3x−2 + x1/2.
Step 2: d/dx[3x−2] = 3(−2)x−3 = −6x−3 = −6/x³.
Step 3: d/dx[x1/2] = 1/2x−1/2 = 1/2√x.
Answer: dy/dx = −6/x³ + 1/2√x
Find the gradient of y = x³ − 4x + 2 at x = 2.
Step 1: Differentiate: y′ = 3x² − 4.
Step 2: Substitute x = 2: y′(2) = 3(4) − 4 = 12 − 4 = 8.
Answer: The gradient at x = 2 is 8.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Differentiate f(x) = x7.
Question 2
Differentiate y = 3x5 + 2x − 9.
Question 3
What is the derivative of a constant, e.g., f(x) = 12?
Question 4
Differentiate y = x−3.
Question 5
Find the gradient of y = 2x³ − x at the point where x = 1.
Key Concepts Summary
- ●Power Rule: d/dx[xn] = nxn−1 for any real number n.
- ●Constant Rule: The derivative of a constant is 0.
- ●Constant Multiple Rule: d/dx[cf(x)] = c · f′(x).
- ●Sum/Difference Rule: Differentiate each term separately.
- ●Rewrite expressions like 1/x and √x as x−1 and x1/2 before differentiating.