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3.1 a – Definition of a Derivative

Finding the ______________________ rate of change is the same as finding the slope of the _____________

line. Another word that describes both of these things is a derivative.

____________________________ = ___________________ = ___________________________ = ________

The derivative is the key to modeling how something changes over time. Derivatives are used to calculate the

velocity, speed, and acceleration of an object. They can find the rate at which the volume of an object is

increasing, how quickly an object is falling, and how fast a plant is growing.

If f’(x) exists, we say f is _____________________________ at the point x. A function that is differentiable at

every point in its domain is a _____________________________ function.

Finding the Derivative in general:

𝑓′(𝑥) = limℎ→0

𝑓(𝑥 + ℎ) − 𝑓(𝑥)

ℎ

Finding the Derivative at a particular point:

𝑓′(𝑎) = lim𝑥→𝑎

𝑓(𝑥) − 𝑓(𝑎)

𝑥 − 𝑎

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3.1 a – Definition of a Derivative

Derivative Notations:

f prime of x:

derivative of y with respect to x:

y prime:

For each function, find f’(x).

1. 𝑓(𝑥) = 𝑥2 − 1 2. 𝑓(𝑥) = √𝑥

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3.1a – Definition of a Derivative

For each function, find dy/dx at the indicated point.

3. 𝑦 = 3𝑥 − 1, 𝑎 = −1 4. 𝑦 =4

𝑥 , 𝑎 = −2

Use the correct formula to find each derivative.

5. 𝐹𝑖𝑛𝑑𝑑𝑦

𝑑𝑥 𝑖𝑓 𝑦 = 2𝑥 + 1. 6. 𝐹𝑖𝑛𝑑

𝑑

𝑑𝑥2𝑥2 𝑎𝑡 𝑥 = 1.