Differentiation Rules

Differentiation Rules Differentiation Rules Rules of differentiation are used to find the derivatives of a given function in calculus. There are some basic rules for differentiation that are used some basic notations as if we want to find the derivative of a given equation that is y = f(x) that shows in the form of f '( x ) = d y/d x or it will be also denoted as y ' = d [ f ( x ) ] / d x .

Rules of Differentiation in calculus are described with some examples given below: ( a ) Differentiation ( derivative ) of a constant function : where f( x ) = c is a function and c is the constant then the derivative of such function is equal to zero .

Example : if f( x ) = 12 then derivative of f( x ) = f '( x ) = 0 . ( b ) Differentiation of a function having power : If the given function is like f ( x ) = x p where p is the power of variable x , is a real number then the derivative of a function is f '( x ) = p x p -1 .

Example : f( a ) = a 3 then the derivative of given function is f '( a ) = 3 a 3 -1 = 3 a 2.

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( c ) Differentiation of a function that multiplied by a constant : If the given function is f( x ) = c g( x ) Then the derivative of this function is f '( x ) = c g '( x ) .

Example : f ( a ) = 3 a 3 then here constant c = 3 and the other function g( x ) = a 3 so by the above rule of differentiation f( x ) = c g'( x ) the derivative of given function is 3 ( 3 * a3-1 ) = 3( 3 a2 ) = 9a 2 . ( d ) Differentiation of the sum of function : If the given function is in the form of f ( x ) = g ( x ) + h ( x ) then the derivative of this is in the form of f '( x ) = g '( x ) + h '( x ) .

Example : If the function f( a ) = a 2 + 4 then here two function g( a )= a 2 and h ( a ) = 4 so , the derivative of this function is f'( a ) = 2 a + 0 = 2 a . ( e ) Differentiation of difference of functions : If the function define as f( x ) = g( x ) - h( x ) then derivative of the given function is f '( x ) = g '( x ) - h '( x ) .

Example : f( a ) = a 3 – a 2 here g( a ) = a 3 and h( a ) = a 2 then the derivative of this given function is f'(a) = 3(a3 – 1) - 2(a2 – 1) = 3a2– 2a . ( f ) Differentiation of product of two function: Function f( a )=g(a).h(a) then the derivative is f'(a)=g(a) h'(a) +h(a) g'(a).

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Application of Differentiation In mathematics, differentiation is a tool which we use in every field like, physics, circuits analysis, dynamics etc.

(1) Derivatives are useful to find out the maxima and minima of any function or any equations.

(2) It met in many engineering and science problems especially to know about the behavior of the moving object.

(3) It is use to find out the tangent and normal of the given curve.

(4) Differentiation are useful to calculating the velocity and acceleration of a moving particle.

(5) We can draw a rough curve or to know about the shape of any given function by curve sketching with the help of differentiation.

(6) In numerical analysis, we can use differentiation in interpolation, in transform calculus we can use the differentiation to solve the given equations.

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