What is a second derivative

First and second derivatives

We will deal with the first and second derivatives of differential calculus in this mathematics article. The focus is initially on the formation of these derivations. In a follow-up article we will look at the benefits of higher derivatives.

In differential calculus one often needs two-way derivatives of a function in order to examine it more closely. In this article we will show you how to make these derivatives. To do this, however, it is absolutely necessary that you know what a derivative actually stands for. You should also already know the basic rules of derivation. If you still have gaps in these areas, the following articles are available to you:

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Higher derivatives

In math lessons, a function is usually given in the form y = f (x). If you derive the function, you get y '(pronounced: Y-dash). If you derive y ', you get y' '(Y-two dash) and so on. The number of "lines" indicates which figure is present. The rules for forming higher derivatives do not change. The following examples show you how this works.

Example 1 (factor rule / power rule):

  • y = 3x3
  • y '= 9x2
  • y '' = 18x

Example 2 (sum rule):

  • y = 5x + 6x3
  • y '= 5 + 18x2
  • y '' = 36x

Example 3 (product rule + factor rule):

  • y = (5x3 -2x) (2x)
  • y '= (15x2 - 2) (2x) + (5x3 - 2x) (2)
  • y '= 30x3 - 4x + 10x3 - 4x
  • y '= 40x3 - 8x
  • y '' = 120x2 - 8

method: Forms the first derivative of the function with the derivation rules. Then, if possible, simplify the function and derive it again. If even higher derivatives (y '' ', y' '' 'etc.) are required, the procedure is applied again. With the knowledge of higher derivatives it is now possible to investigate extreme value problems.

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