Second and higher derivatives.





Think of a function y=f(x) , and let y=f(x) be a differentiable function.

Then you can differentiate the function f(x) with respect to x or find derivative of f(x)=  \frac{d}{dx}f(x) which is called the first derivative of the function f(x). Let the first derivative of the function be “y” and now if you again differentiate y with respect to “x” then the result is called second derivative of function f(x) you can again find it’s derivative and that’s called the third derivative of function f(x) and similarly you can differentiate again to find fourth , fifth , sixth…… derivative.

Mathematically you can denote second derivative as:
\frac{d^2}{dx^2}\left(f(x)\right)

The third derivative is denoted by:

\frac{d^3}{dx^3}\left(f(x)\right)

And you can also denote fourth , fifth and all higher derivative by changing the power of “d” and “x” to the respective number.

You can find the second , third , fourth and higher derivative using the following formula:

\frac{d^2}{dx^2}\left(f(x)\right)=\frac{d}{dx}\left(\frac{d}{dx}f(x)\right)

\frac{d^3}{dx^3}\left(f(x)\right)=\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}f(x)\right)\right)
And so on…….

Example of Second and Higher Derivative:

Let us find the second , third , fourth and higher derivatives of the function:

f(x)=5x4+4x3-18

Solution:

The first derivative of the function=
\frac{d}{dx}\left(5x^4+4x^3-18\right) = 20x^3+12x^2

The Second derivative of the function=
\frac{d^2}{dx^2}\left(5x^4+4x^3-18\right) = \frac{d}{dx}\left(20x^3+12x^2\right)=60x^2+24x

The third derivative of the function=
\frac{d^3}{dx^3}\left(5x^4+4x^3-18\right) = \frac{d}{dx}\left(60x^2+24x\right)=120x+24

The fourth derivative of the function=
\frac{d^4}{dx^4}\left(5x^4+4x^3-18\right) = \frac{d}{dx}\left(120x+24\right)=120

The fifth derivative of the function=
\frac{d^5}{dx^5}\left(5x^4+4x^3-18\right) = \frac{d}{dx}120=0

And as fifth derivative of the function f(x) is 0 sixth , seventh , eighth…. and all the higher derivative of the function will be 0.



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