Differentiation

Definition of Derivative

The notation around differentiation can sometimes be confusing. The important thing to remember is that we take the derivative of a ~{function}, and that the result is another ~{function}.

As an example consider the sinus function, which we usually write something like: _ sin ( ~x ). Here the ~x is the argument of the function, and just represents any real value. The function can be thought of as a machine which takes one real number ( and for the time being we are only talking about functions of a single real-value ) and the result is another real number.

In simplistic terms, providing everything is nice an continuous and so on, the derivative of the function for a particular number is the gradient of the tangent of the graph of the function at that number. So the derivative of sin ( ~x ) at &pi./2 is 0, and the derivative at the point 0 is 1. We can work this out for any real value, so what we have is another function, in this case it is of course the cosinus, _ cos ( ~x ).

If _ ~f ( ~x ) _ is a function of ~x, then its derivative will be denoted _ D~f , _ and is defined

D~f ( ~x ) _ #:= _ ~f_~x* ( ~x ) , _ where _ ~f_~x* ( ~a ) _ = _ ( ~f ( ~a ) - ~f ( ~x ) ) ./ ( ~a - ~x ) ,
[ Providing &exist. continuous _ ~f_~x* _ &forall. ~x , _ such that _ ( ~a - ~x ) ~f_~x* ( ~a ) _ = _ ( ~f ( ~a ) - ~f ( ~x ) ) . ]

The derivative of ~f is more ususally denoted

fract{d ~f,d~x} , _ _ or _ _ ~f #~'

[ d~f/d~x _ can be thought of as _ d/d~x ( ~f ) , _ i.e _ d/d~x is the ~{operator} _ D _ which operates on the function _ ~f.]

 

 

Properties of Derivatives

~f and ~g are functions of ~x:

fract{d ( ~c ~f ),d~x} _ = _ ~c fract{ d ~f _ ,d~x} , _ ( ~c constant ) fract{d ( ~f + ~g ),d~x} _ = _ fract{d ~f,d~x} + fract{d ~g,d~x} fract{d ( ~f ~g ),d~x} _ = _ ~{f }fract{d ~g,d~x} + ~g fract{d ~f,d~x} fract{d ( ~f ./ ~g ),d~x} _ = _ ( ~g fract{d ~f,d~x} - ~{f }fract{d ~g,d~x} ) _ ./ _ ~g ^2 #~{function of a function}: fract{d ( ~h ( ~g ) ),d~x} _ = _ fract{d ~h,d~g} fract{d ~g,d~x} , _ where _ ~f ( ~x ) _ = _ ~h ( ~g ( ~x ) ) #~{inverse function}: ~x = ~f ( ~y ) , _ then _ fract{d~y,d~x} _ = _ fract{1,d ~f ./ d~y}

Powers of x

1. ~f ( ~x ) _ = _ ~c , _ ( ~c constant ) , _ D~f ( ~x ) _ = _ 0
2. ~f ( ~x ) _ = _ ~x , _ ~f_~x* ( &theta. ) _ = 1 _ _ => _ D~f ( ~x ) _ = _ 1
3. ~f ( ~x ) _ = _ ~x^2 , _ &theta.^2 - ~x^2 _ = _ ( &theta. - ~x ) ( &theta. + ~x ) _ _ => _ ~f_~x* ( &theta. ) _ = _ &theta. + ~x _ _ => _ D~f ( ~x ) _ = _ 2~x
4. ~f ( ~x ) _ = _ ~x^~n , _ D~f ( ~x ) _ = _ ~n ~x^{~n - 1} _ ( ~n &in. Z^+ )
   Proof by induction. True for ~n = 1 , _ suppose D ( ~x^{~n - 1} ) _ = _ (~n - 1)~x^{~n - 2} .
   then _ D ( ~x^~n ) _ = _ D ( ~x . ~x^{~n - 1} ) _ = _ ~x (~n - 1)~x^{~n - 2} + 1 . ~x^{~n - 1} _ (prop. 3) _ = _ ~n ~x^{~n - 1}.
5. ~f ( ~x ) _ = _ 1 ./ ~x . ( 1/ ~a ) - ( 1 ./ ~x ) = ( ~x - ~a ) ./ ~a~x _ , _ => _ ~f_~x* ( ~a ) _ = _ -1 ./ ~a~x _ => _ D~f ( ~x ) _ = _ -1 ./ ~x^2
6. ~f ( ~x ) _ = _ ~x^{-~n} , _ D~f ( ~x ) _ = _ -~n ~x^{-(~n + 1)} _ ( ~n &in. Z^+ )
   Proof by induction. True for ~n = 1 _ (ex. 5) , _ suppose D ( ~x^{-(~n - 1)} ) _ = _ -(~n - 1)~x^{-~n} .
   then _ D ( ~x^{-~n} ) _ = _ D ( ( ~x^~n )^{-1} ) _ = _ -1 ( ~x^~n ) ^{-2} . ~n ~x^{~n -1} _ (prop. 5) _ = _ -~n ~x^{-(~n + 1)} .
7. ~f ( ~x ) _ = _ ~x^{1/~n} _ ( ~n &in. Z^+ ) , _ ~y = ~x^{1/~n} = ~f ( ~x ) _ <=> _ ~x = ~y^~n = ~f ^{-1}( ~y ) .
   D~f ( ~x ) _ = _ 1 ./ D(~f ^{-1}) ( ~f ( ~x ) ) _ = _ 1 ./ ( ~n ~y^{~n - 1} ) _ = _ ~y ./ ( ~n ~y^~n ) _ = _ ~x^{1/~n} ./ ~n~x _ = _ ( 1/~n ) ~x^{( 1/~n ) - 1}
8. Combining the above results, (abusing the notation slightly) we see that

   fract{d ~x^{~q},d~x} _ = _ ~q ~x^{~q - 1} , _ ( ~q &in. Q )

Trigonometrical Functions

1. ~f ( ~x ) _ = _ sin ~x . _ D~f ( ~x ) _ = _ cos ~x .
   sin ~a - sin ~x _ = _ sin ( (~a - ~x) + ~x ) - sin ~x _ = _ sin (~a - ~x) cos ~x + ( cos (~a - ~x) - 1 )sin ~x
   => _ ~f_~x* ( ~a ) = \{ (sin (~a - ~x) cos ~x + ( cos (~a - ~x) - 1 )sin ~x \} ./ ( ~a - ~x ) . This is continuous as _ sin (~a - ~x) ./ ( ~a - ~x ) -> 1 _ and _ cos (~a - ~x) ./ ( ~a - ~x ) -> 0 _ as _ ~a -> ~x . _ So _ D~f ( ~x ) _ = _ ~f_~x* ( ~x ) _ = _ cos ~x .
2. ~f ( ~x ) _ = _ cos ~x . _ D~f ( ~x ) _ = _ - sin ~x .
   cos ~a - cos ~x _ = _ cos ( (~a - ~x) + ~x ) - cos ~x _ = _ cos (~a - ~x) cos ~x - ( sin (~a - ~x) sin ~x ) - cos ~x
   = _ (cos (~a - ~x) - 1 ) cos ~x - ( sin (~a - ~x) sin ~x )
   => _ ~f_~x* ( ~a ) = \{ (cos (~a - ~x) - 1 ) cos ~x - ( sin (~a - ~x) sin ~x ) \} ./ ( ~a - ~x ) . _ This is continuous as _ sin (~a - ~x) ./ ( ~a - ~x ) -> 1 _ and _ cos (~a - ~x) ./ ( ~a - ~x ) -> 0 _ as _ ~a -> ~x . _ So _ D~f ( ~x ) _ = _ ~f_~x* ( ~x ) _ = _ - sin ~x .
3. ~f ( ~x ) _ = _ tan ~x . _ D~f ( ~x ) _ = _ 1 ./ cos^2 ~x _ = _ 1 + tan^2 ~x .
   tan ~x = sin ~x ./ cos ~x , _ so by prop. 4 , _ D~f ( ~x ) _ = _ ( cos ~x ( cos ~x ) - sin ~x ( - sin ~x ) ) ./ cos^2 ~x
   = _ ( cos^2 ~x + sin^2 ~x ) ./ cos^2 ~x _ = _ 1 ./ cos^2 ~x _ = _ sec^2 ~x _ _ ( or _ 1 + tan^2 ~x )

   fract{d sin ~x,d~x} _ _ = _ _ cos ~x fract{d cos ~x,d~x} _ _ = _ _ - sin ~x fract{d tan ~x,d~x} _ _ = _ _ sec^2 ~x

4. D( cosec ~x ) _ = _ D( (sin ~x)^{-1} ) _ = _ - (sin ~x)^{-2} cos ~x _ = _ - cosec ~x cot ~x ,
   and similarly for the other trigonometrical reciprocals:

   fract{d cosec ~x,d~x} _ _ = _ _ - cosec ~x cot ~x fract{d sec ~x,d~x} _ _ = _ _ - sec ~x tan ~x fract{d cot ~x,d~x} _ _ = _ _ - cosec^2 ~x

5. ~f ( ~x ) _ = _ sin^{-1}~x _ = _ arc sin ~x . _ ~y = sin^{-1}~x _ <=> _ ~x = sin ~y .
   By property 6 , _ D~f ( ~x ) _ = _ 1 ./ D(~f ^{-1}) ( ~f ( ~x ) ) _ = _ 1 ./ cos ~y _ = _ 1 ./ &sqrt.${1 - sin&powtwo. ~y} _ = _ 1 ./ &sqrt.${1 - ~x&powtwo.} .
6. ~f ( ~x ) _ = _ cos^{-1}~x _ = _ arc cos ~x . _ ~y = cos^{-1}~x _ <=> _ ~x = cos ~y .
   By property 6 , _ D~f ( ~x ) _ = _ 1 ./ D(~f ^{-1}) ( ~f ( ~x ) ) _ = _ - 1 ./ sin ~y _ = _ - 1 ./ &sqrt.${1 - cos&powtwo. ~y} _ = _ - 1 ./ &sqrt.${1 - ~x&powtwo.} .
7. ~f ( ~x ) _ = _ tan^{-1}~x _ = _ arc tan ~x . _ ~y = tan^{-1}~x _ <=> _ ~x = tan ~y .
   By property 6 , _ D~f ( ~x ) _ = _ 1 ./ D(~f ^{-1}) ( ~f ( ~x ) ) _ = _ 1 ./ sec^2 ~y _ = _ cos^2 ~y _ = _ cos^2 ~y ./ ( cos^2 ~y + sin^2 ~y ) _
   = _ 1 ./ ( 1 + tan^2 ~y ) _ = _ 1 ./ ( 1 + ~x^2 ) .

   fract{d sin^{-1}~x,d~x} _ _ = _ _ fract{1,&sqrt.${1 - ~x&powtwo.}} fract{d cos^{-1}~x,d~x} _ _ = _ _ - fract{1,&sqrt.${1 - ~x&powtwo.}} fract{d tan^{-1}~x,d~x} _ _ = _ _ fract{1,( 1 + ~x^2 )}

8. D( sin^{-1}( ~x/~a ) ) _ = _ 1 ./ ( ~a &sqrt.${1 - ( ~x/~a )&powtwo.} ) _ = _ 1 ./ &sqrt.${~a&powtwo. - ~x&powtwo.} . _ _ Similarly the other functions:

   fract{d sin^{-1}( ~x/~a ),d~x} _ _ = _ _ fract{1,&sqrt.${~a&powtwo. - ~x&powtwo.}} fract{d cos^{-1}( ~x/~a ),d~x} _ _ = _ _ - fract{1,&sqrt.${~a&powtwo. - ~x&powtwo.}} fract{d tan^{-1}( ~x/~a ),d~x} _ _ = _ _ fract{1, ~a^2 + ~x^2 }

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