1. Mathematics
2. Calculus and Analysis
3. Real Analysis
4. Derivatives
5. 148848

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Key points:
The derivative of at is the instantaneous rate of change of and it is denoted by .

If we let , we obtain an equivalent formula

that is at times more convenient to use in problems. A function for which the exists is called differentiable at . A function is differentiable on an interval if it is differentiable at each number in that interval.
Given we define a new function . This is the derivative function. The derivative describes how quickly is changing. Other notations for the derivative function are (where ), , .
Interact (use left and right arrow keys) with this (quicktime) animation that illustrates a function and its derivative. Before you play it try to sketch on a piece of paper.
The derivative of the derivative of is denoted by: or sometimes by .

1. Use the definition of derivative to calculate where .
2. Use the definition of derivative to calculate where .
3. Use the definition of derivative to calculate where .
4. Use the definition of derivative to find .