Definition of the Derivative
Instantaneous rate of change as a limit of secant slopes.
Key idea
Instantaneous rate of change as a limit of secant slopes.
In depth
f′(a) = lim (f(a + h) − f(a))/h as h → 0: the limit of secant slopes becomes the slope of the tangent line. It measures instantaneous rate of change.
Worked example
For f(x) = x², (x + h)² − x² = 2xh + h², divide by h → 2x + h → 2x.
Formula
Real-world uses
- Defining instantaneous speed, growth and marginal cost in physics and economics
Common mistake
Substituting h = 0 before simplifying, which gives 0/0.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Definition of the Derivative?
Instantaneous rate of change as a limit of secant slopes. It is a topic at the advanced level in Calculus.
What is the formula for Definition of the Derivative?
The key formula is f′(x) = lim_{h→0} (f(x+h)−f(x))/h.
Where is Definition of the Derivative used in real life?
Real-world examples: Defining instantaneous speed, growth and marginal cost in physics and economics.
What should I learn before Definition of the Derivative?
Make sure you are comfortable with Average Rate of Change, the previous topic in Calculus.
What is a common mistake with Definition of the Derivative?
Substituting h = 0 before simplifying, which gives 0/0.
How can I practise Definition of the Derivative?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Definition of the Derivative?
Move on to Tangent Lines next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.