Derivative

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The functional derivatives represents a minute modification in the function with respect to one of its variables. The "simple" derivative of a function “f” with respect to a variable x is denoted either f(x) or

 (df)/(dx)…….(1)

It is often written in-line as df/dx. When derivatives are taken relating to time, they are being denoted using Newton's overdot (A single dot above x) note for fluxions,

 (dx)/(dt)=x^..   ……..(2)

When any derivatives are taken n times, the notation f^(n) (x) or we can represent as:

 (d^nf)/(dx^n) …….(3)

There are some important rules for computing derivatives of definite combinations of functions.  Derivatives of sums are exactly equal to the sum of derivatives, so that

[f (x)+…..+h(x)]’ = f’ (x)+…..+h’ (x)
 

f¹(x) is the derivative of f(x) which is defined as
Example 1: Find the derivative of f(x) = x² -8x +12. Find the derivative by using the definition of derivative.
Answer: 1st Method
Formulae:
ddx (xn) = n x(n-1)
ddx (a) = 0, here a is constant
These formulae can also be used in order to find the derivatives.
Example 2: Find the derivative of f(x) = x² -8x +12
Answer: 2nd method
ddx (xn) = n x(n-1)
ddx (x2) = 2 x2-1 = 2x
ddx (-8x) = -8 ddx (x)= -8.1.x1-1 = -8
ddx (12) = 0
f¹(x) = 2x - 8

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