9. Higher Derivatives
by M. Bourne
We can continue to find the derivatives of a derivative. We find the
- second derivative by taking the derivative of the first derivative,
- third derivative by taking the derivative of the second derivative... etc
Example
If 
, then what are the higher derivatives?
Here are the LiveMath answers for this example.
Application - Acceleration
We saw before that acceleration is the rate of change of velocity:
But we also know that velocity is the rate of change of displacement:
So it follows that the second derivative of displacement will give us acceleration:
Example
If the displacement (in metres) at time t (in seconds) of an object is given by
s = 4t3 + 7t2 - 2t,
find the acceleration at time t = 10.
Solution:
s = 4t3 + 7t2 - 2t
At t = 10, the acceleration will be
a = 24(10) + 14 = 254 ms-2.
Higher Derivatives of Implicit Functions
Example
a. Find the second derivative of the implicit function xy + y2 = 4.
b. Find the value of the second derivative of the implicit function in part (a) when x = 2, where y > 0.
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