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 1

If `y=x^5+3x^3-2x+7`, then what are the higher derivatives?

Application - Acceleration

We saw before that acceleration is the rate of change of velocity:

`a=(dv)/(dt)`

But we also know that velocity is the rate of change of displacement:

`v=(ds)/(dt)`

So it follows that the second derivative of displacement will give us acceleration:

`a=(d^2s)/(dt^2)`

Example 2

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`.

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Higher Derivatives of Implicit Functions

Example 3

(The Answers for these two questions contain short video explanations.)

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.