From average to instantaneous
The average rate of change of f over [a, a + h] is the slope of a secant line: (f(a + h) - f(a))/h. As h shrinks toward 0 the secant line swings into the tangent line. The slope of that tangent is the derivative at a:
f'(a) = lim(h -> 0) (f(a + h) - f(a))/h
Replace the fixed a with a variable x and you get the derivative function f'(x). Other notations: dy/dx and df/dx. The derivative measures the instantaneous rate of change, such as speed at one moment.
Computing from the definition
The plan never changes: write the difference quotient, simplify until h cancels from the bottom, then let h go to 0.
Worked example
Let f(x) = x^2 + 3x. Find f'(2).
- f(2 + h) = (2 + h)^2 + 3(2 + h) = 4 + 4h + h^2 + 6 + 3h = 10 + 7h + h^2.
- f(2) = 4 + 6 = 10.
- The difference quotient is (7h + h^2)/h = 7 + h.
- As h approaches 0, this approaches 7. So f'(2) = 7.
Check: at h = 0.0001 the quotient is about 7.0001, which agrees.
Tangent line: at x = 2 the point is (2, 10) and the slope is 7, so the tangent is y - 10 = 7(x - 2), which is y = 7x - 4.
Differentiable means continuous
If f is differentiable at a, then f is continuous at a. The reverse is false. The function |x| is continuous at 0, but its graph has a sharp corner there: the secant slopes are -1 from the left and +1 from the right, so the derivative does not exist. Vertical tangents and jumps also destroy differentiability.
Reading derivatives
Where f'(x) > 0 the function is rising, where f'(x) < 0 it is falling, and where f'(x) = 0 the tangent is horizontal. If s(t) is position in metres at time t seconds, then s'(t) is velocity in metres per second.
Common mistakes
- Leaving the h in the bottom: always simplify first, and never plug in h = 0 too early (that gives 0/0).
- Squaring wrongly: (a + h)^2 is a^2 + 2ah + h^2, not a^2 + h^2.
- Confusing the value f(a) with the slope f'(a).
- Believing continuity guarantees a derivative.