Understanding Derivative Rules
Learn the key rules for finding derivatives of various functions.
Derivatives measure the rate of change of a function. There are several rules that help us find derivatives efficiently.
Power Rule
If f(x) = xn, then f'(x) = n·xn-1
Examples:
- If f(x) = x2, then f'(x) = 2x
- If f(x) = x3, then f'(x) = 3x2
- If f(x) = x-1, then f'(x) = -x-2
Key Points:
- Works for any real number n
- Simplifies finding derivatives of polynomials
- Can be combined with other rules
Product Rule
If f(x) = g(x)·h(x), then f'(x) = g'(x)·h(x) + g(x)·h'(x)
Examples:
- If f(x) = x·sin(x), then f'(x) = 1·sin(x) + x·cos(x)
- If f(x) = x2·ex, then f'(x) = 2x·ex + x2·ex
Key Points:
- Used when multiplying two functions
- Remember: "First times derivative of second plus second times derivative of first"
Chain Rule
If f(x) = g(h(x)), then f'(x) = g'(h(x))·h'(x)
Examples:
- If f(x) = sin(x2), then f'(x) = cos(x2)·2x
- If f(x) = (x3 + 1)4, then f'(x) = 4(x3 + 1)3·3x2
Key Points:
- Used for composite functions (functions of functions)
- Derivative of outer function evaluated at inner function, times derivative of inner function