Rules of Differentiation
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 <sup>p</sup> where p is the power of variable x , is a real number then the derivative of a function is f '( x ) = p x <sup>p -1 </sup>.
Example : f( a ) = a <sup>3</sup> then the derivative of given function is f '( a ) = 3 a <sup>3 -1 </sup>= 3 a <sup>2</sup>.
( 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 <sup>3</sup> then here constant c = 3 and the other function g( x ) = a <sup>3</sup>
so by the above rule of differentiation f( x ) = c g'( x ) the derivative of given function is
3 ( 3 * a<sup>3-1 </sup>) = 3( 3 a<sup>2 </sup>) = 9a <sup>2 </sup>.
( 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 <sup>2</sup> + 4 then here two function g( a )= a <sup>2</sup> 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 <sup>3</sup> €" a <sup>2 </sup>here g( a ) = a <sup>3</sup> and h( a ) = a <sup>2</sup> then the derivative of this given function is
f'(a) = 3(a<sup>3 €" 1</sup>) - 2(a<sup>2 €" 1</sup>) = 3a<sup>2</sup>€" 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).
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 <sup>p</sup> where p is the power of variable x , is a real number then the derivative of a function is f '( x ) = p x <sup>p -1 </sup>.
Example : f( a ) = a <sup>3</sup> then the derivative of given function is f '( a ) = 3 a <sup>3 -1 </sup>= 3 a <sup>2</sup>.
( 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 <sup>3</sup> then here constant c = 3 and the other function g( x ) = a <sup>3</sup>
so by the above rule of differentiation f( x ) = c g'( x ) the derivative of given function is
3 ( 3 * a<sup>3-1 </sup>) = 3( 3 a<sup>2 </sup>) = 9a <sup>2 </sup>.
( 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 <sup>2</sup> + 4 then here two function g( a )= a <sup>2</sup> 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 <sup>3</sup> €" a <sup>2 </sup>here g( a ) = a <sup>3</sup> and h( a ) = a <sup>2</sup> then the derivative of this given function is
f'(a) = 3(a<sup>3 €" 1</sup>) - 2(a<sup>2 €" 1</sup>) = 3a<sup>2</sup>€" 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).