# Math 22 Higher-Order Derivative

## Higher-Order Derivatives

The "standard" derivative is called the first derivative of . The derivative of is the second derivative of, denoted by By continuing this process, we obtainhigher-order derivativeof .

Note: The 3rd derivative of is . However, we simply denote the derivative as for

**Example**: Find the first four derivative of

**1)**

Solution: |
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**2)**

Solution: |
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It is better to rewrite |

Then, |

## Acceleration

If is the position function, then is the velocity function and is the acceleration function.

**Word-Problem Example**: A ball is thrown upward from the top of a -foot cliff. The initial velocity of the ball is feet per second. The position function is where is measured in seconds. Find the height, velocity, and acceleration of the ball at

Solution: |
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(Position function) |

(Velocity function) |

(Acceleration function) |

So, when , from the functions above, we can have: |

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