Second Order Derivatives
Second Order Derivative tells us about the rate of change of derivative of a function. It is used to find the maximum and minimum value of a function. It also gives the optimal solution of any equation including functions. We find the second order derivative of any function by further derivating the function.
Let’s say we have a function f(x).
y = f(x)
dy/dx = f'(x)
If f'(x) is differentiable function, we can differentiate it again to get a second-order derivative. It is denoted by,
d/dx{f'(x)} = d2y/dx2
Second Order Derivative are represented as, f”(x).
Example: Given y = x/(x2 + 1). Find the value of second derivative at x = 1
Solution:
Given,
y = x/(x2 + 1)
[Tex]\frac{d}{dx}(\frac{g(x)}{h(x)}) = \frac{g(x)h'(x) – h(x)g'(x)}{h(x)^2} [/Tex]
y’ = [Tex]\frac{(x^2+1) – x(2x)}{(x^2 + 1)^2} [/Tex]
y’ = [Tex]\frac{1 – x^2}{(x^2 + 1)^2} [/Tex]
Now we can differentiate it again to get the second derivative.
y”=[Tex]\frac{(x^2 + 1)^2(-2x) – (1 – 2x^2)(2(x^2+1)2x)}{(x^2 + 1)^4} [/Tex]
At x = 1
y” = [Tex]\frac{(1^2 + 1)^2(-2) – (1 – 21^2)(2(1^2+1)2)}{(1^2 + 1)^4} [/Tex]
y” = [Tex]\frac{(2)^2(-2) – (1 – 2)(2(2)2)}{(2)^4} [/Tex]
y” = [Tex]\frac{-8 + 8}{(2)^4} [/Tex]
y” = 0
Thus, second order derivative of y = x/(x2 + 1) at x = 1 is, y” = 0.
Higher Order Derivatives
Higher Order Derivatives are the second, third, or further derivative of the function, i.e. differentiating a function multiple times results in a higher order derivative. Suppose we have a function f(x) then its differentiation is f'(x) which is a first-order derivative. Then its differentiation again f”(x) is called the second order derivative. This is the Higher Order Derivative and then differentiating the second order derivative again results in the third order derivative i.e. f”'(x) is the third derivative of the function f(x) and then higher derivatives are further calculated.
These Higher Order Derivatives are used for various purposes they are used to find the maxima and minima of the function, it is also used for finding the optimal solution of a function, etc.
In this article, we will learn about, Higher Order Derivatives definition, Second Order derivative, Third Order derivative, examples, and others in detail.
Table of Content
- Higher Order Derivatives Definition
- Second Order Derivatives
- Third Order Derivative
- Higher-Order Derivative in Parametric Form
- Application of Higher Order Derivative
- Examples
- FAQs
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