Short answer
The derivative at x = a is f'(a) = lim h->0 [f(a+h)-f(a)]/h. It measures instantaneous change by shrinking the secant interval to zero.
- Derivative comes from a limit of average rate of change.
- Keep the difference quotient unsimplified until common factors appear.
- A derivative value is a slope and also a rate with units.
Build the difference quotient correctly
Start with f(a+h) and f(a), then divide by h. Expand carefully and combine like terms before canceling h.
Only after simplification should you take h to 0. This is the key move that turns average rate into instantaneous rate.
Interpret derivative values in context
A positive derivative means increasing at that point, a negative derivative means decreasing, and zero means horizontal tangent behavior at that instant.
Units matter: if f is meters and x is seconds, then f' is meters per second.
- Use clear variable labels before computing.
- Keep h symbolic until cancellation is complete.
- Report both numeric slope and meaning in words.
Find the derivative of f(x) = x^2 at x = 4
Use the limit definition, not shortcut rules.
- Write f'(4) = lim h->0 [(4+h)^2 - 16]/h.
- Expand numerator: 16 + 8h + h^2 - 16 = 8h + h^2.
- Factor h, cancel, then take limit of 8 + h as h->0.
Result: f'(4) = 8, so the tangent slope at x = 4 is 8.
Common mistakes
- Using x instead of h and losing track of the target point.
- Canceling h before factoring h from every term.
- Treating 0/0 as the final answer.
- Reporting slope without units or interpretation.
Try one
What does f'(3) represent in plain language?
It is the instantaneous rate of change of f with respect to x when x = 3.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.