The derivative and the derivative function
Derivative and common derivatives
From f'(a) to the function f'
If f is differentiable at every point in an interval I, we can define the derivative function f', which associates the number f'(x) with every x in I. This allows us to study f without having to resort to a limit every time.
Derivatives of common functions
| Function f | Derivative f' | Domain |
|---|---|---|
| f(x) = k (constant) | f'(x) = 0 | R |
| f(x) = x | f'(x) = 1 | R |
| f(x) = x² | f'(x) = 2x | R |
| f(x) = x^n (n an integer ≥ 1) | f'(x) = n*x^(n-1) | R |
| f(x) = 1/x | f'(x) = -1/x² | x ≠ 0 |
| f(x) = √x | f'(x) = 1/(2√x) | x > 0 |
Operations on derivatives
If u and v are two differentiable functions on an interval I and k is a real number:
- (u + v)' = u' + v'
- (ku)' = ku'
- (u*v)' = u'v + uv'
- (u/v)' = (u'v - uv') / v², provided v is non-zero on I
Example (sum): f(x) = 3x² - 5x + 1, so f'(x) = 6x - 5.
Example (product): g(x) = (2x + 1)(x - 3). g'(x) = 2*(x-3) + (2x+1)*1 = (2x-6) + (2x+1) = 4x - 5
Common pitfall
A very common mistake is to believe that (u*v)’ = u’*v’. This is incorrect! Similarly, (u/v)’ is not u’/v’. You must always apply the exact formulas given above, clearly identifying u and v before differentiating.