The Power Rule
The Power Rule
The most used derivative rule
The power rule is the workhorse of calculus. You'll use it constantly. Good news: it's simple!
Power Rule
Power Rule
d/dx[x^n] = n*x^(n-1)
Bring the exponent down as a coefficient, then reduce the exponent by 1
Works for ANY Exponent
The power rule works for:
- Positive integers: x^5
- Negative integers: x^(-3) = 1/x^3
- Fractions: x^(1/2) = sqrt(x)
- Any real number: x^pi
Basic Power Rule
Easy
Find d/dx[x^7]
1
Identify n
n = 7
2
Apply power rule
d/dx[x^7] = 7x^(7-1) = 7x^6
Answer:
7x^6
Power Rule with Roots
Medium
Find d/dx[sqrt(x)]
1
Rewrite as power
sqrt(x) = x^(1/2)
2
Apply power rule
d/dx[x^(1/2)] = (1/2)x^(1/2-1) = (1/2)x^(-1/2)
3
Simplify
= 1/(2sqrt(x))
Answer:
1/(2sqrt(x))
Practice
1. d/dx[x^10] = ?
2. d/dx[1/x^3] = ?
1/x^3 = x^(-3). Power rule: -3x^(-4) = -3/x^4
Key Takeaways
Power Rule: d/dx[x^n] = nx^(n-1)
Bring exponent down, subtract 1 from exponent
Works for ALL real exponents (positive, negative, fractions)
Rewrite roots and fractions as powers first