What is the formal definition of a derivative?

Formal Definition of a Derivative

The derivative of a function at a point measures how the function value changes as its input changes. Formally, if f(x) is a function, the derivative at x = a is defined as:

``
f'(a) = lim_(h→0) [f(a+h) - f(a)] / h
`

• Calculate the average rate of change between x = a and x = a+h
• Let h get closer and closer to 0

Quick Example

For f(x) = x^2, at x = 3:

`
f'(3) = lim_(h→0) [(3+h)^2 - 3^2] / h
= lim_(h→0) [9 + 6h + h^2 - 9] / h
= lim_(h→0) [6h + h^2] / h
= lim_(h→0) 6 + h
= 6
`
So, the derivative at x = 3 is 6`.

Takeaways

• The derivative formally describes the instant rate of change of a function using a limit.
• It's foundational for calculus, measuring how functions change at specific points.