DIFFERENTIATION OF VARIABLE POWER FUNCTIONS USING PRODUCT RULE

Let us learn and remember  easiest method of differentiation, differentiate,differentiation formula of the functions which involve variable power in it in an easy and short cut manners.

Till Date we have learn to differentiate this question by taking log on both sides and then differentiate.

This is very long process to differentiate this types of functions .
But today we shall learn a different and an easiest method to differentiate such type of functions.
      

then assume this function as [ log (Base) ]× [ Power ] then use product rule of differentiation and place the given function in front of the result so obtained.

f '(x) = [ log (Base) ] d/dx [ Power ] + [ Power ] d/dx [ log (Base) ]

Question  Differentiate f(x) = (cos x )^{sin x}

Consider    h(x) = (log cos x) ×(sin x)

then it derivative will be 

 f '(x) = f(x) Diff  (h(x))

⇒ f '(x) = f(x) [(log cos x) . Diff (sin x) + sin x Diff (log cos x)]

Therefore  f '(x) = f(x) [(log cos x) . cos x + sin x (-sin x ) /cos x)]

Therefore  f '(x) = (cos x )^{sin x} [(log cos x) . cos x - sin x .tan x]

Question : How to solve this f(x) = x ^{sin x}  

First Assume base x as Log x as 1st function and power function as 2nd function, then apply Product rule of differentiation,and place f(x) in front of the result so obtained.
d/dx { log x . sin x }= ( log x) d/dx ( sin x ) + ( sin x ) d/dx ( log x)

= log x . cos x + sin x . (1/x)                            
                     
Now put f(x) in front of this result and that will be derivative of the f(x).Hence f ' (x) = x ^{sin x} { log x . cos x + sin x . (1/x) }

Question : Differentiate w.r.t. 'x'

 f(x) = cos x ^{sin x} + (sin x) ^x 

Let  f(x) = g(x) +h(x)
Then   f '(x) =g'(x) +h'(x)


Just place cos x ^{sin x}  in front of derivative of {(log cos x) . (sin x) } + place  (sin x) ^x  in front of derivative of { ( log sin x) . ( x) },

So Answer will be g ' (x) = cos x ^{sin x}   { log cos x . cos x +  sin x . (- sin x ) /cos x } 

and h'(x) =  (sin x) ^x { log sin x . (1/x) + x . cos x }

Similarly derivative of h(x) = (sin x) ^x    in one step can also be written as
h '(x) = (sin x) ^x [ log sin x × 1+ x . cos x/sin x ]

Question : Differentiate f (x) = e ^{sin x} 

 then using short cut method , 

f '(x) = e ^{sin x}  [ log e . d/dx{sin x} + sin x d/dx{log  e}]

f '(x) = e ^{sin x}  [ log e × cos x ] ,

Because derivative of log e is zero 

Question : Differentiate f (x) = a ^{sin x} 

If f (x) = a ^{sin x} 

then using short cut method , 

f '(x) =a ^{sin x} [ log a . d/dx{sin x} + sin x d/dx{log  a}]

f '(x) = a ^{sin x} [ log a . cos x ] ,

Because derivative of log a is zero 

Question : Differentiate f (x) = x ^{sin x + cos x} 

If f (x) = x ^{sin x + cos x} 

then using short cut method , 

f '(x) = x ^{sin x + cos x} [ log x . d/dx{sin x + cos x } + {sin x + cos x }d/dx{log x}]

f '(x) = x ^{sin x + cos x} [ log x  . {cos x - sin x} + {sin x+cos x }.{1/x} ] 

One more shortcut for differentiation you can use

            

Conclusion

Thanks for devoting your valuable time for this post  differentiate or differentiation formula "Product Rule to Differentiate  Logarithmic Function of variable powers"  of my blog . If you liked this this blog/post,  Do Follow me on my blog and share this post with your friends . We shall meet again   in next post with solutions of most interesting and mind blowing mathematics problems ,till then Good Bye.

Read More

HOW TO FIND THE TRANSPOSE OF MATRIX WITH AN EASY METHOD

In this post we are going to learn "what is transpose of matrix, calculate  transpose ,define transpose, meaning of transpose " .  As it is clear from its name  transpose means trans + pose i. e. transfer + position ( transfer of position ) ,  the transpose   of any matrix is obtained by transfer of Rows into Columns And vice versa. i.e transforming 1st row to 1st column and transforming 2nd row to 2nd column and so on for 3rd ,4th and 5th rows and columns. This is very easy and interesting topic in matrices and determinants.

 To find the transpose of the matrix 

1^st of all  shift all the elements which are in 1st row to 1^st column as

5

5

2
,then shift the elements which are in      2nd row to 2^nd column as  

-1

-3

 7

similarly  shift all the elements which are in 3rd row to 3^rd column as

4

2

8

And the matrix so obtained is the transpose matrix. We can  check that same colour row have been  transformed to same colour Column
        

Now we shall take one  example to find the transpose matrix

Here Given matrix have 3 rows and 4 columns .It means we shall have 4 rows and 3  columns  in transpose matrix.

1st shift all the elements which are in 1st row to 1^st column as

5
8               
6
-4

shift all the elements which are in 2nd row to 2nd column as

3
5
8
3

shift all the elements which are in 3rd row to 3rd column as 

-3
8
-7
6

so shifting the corresponding Rows into Corresponding columns. We can  check that same colour rows are transformed to same colour columns.

Now we shall take one more  example to find the transpose of matrix .Here Given matrix have 3 rows and 4 columns,It means  we shall have 4 rows and  3 columns in transpose of that matrix.

1st shift all the elements which are in 1st row to 1^st column as

 7

-1

-2

 5

shift all the elements which are in 2nd row to 2nd column as

4

4

3

3

shift all the elements which are in 3rd row to 3rd column as 

  8

 9

 6

-1

so shifting the corresponding Rows into Corresponding columns. We can that check same colour rows are transformed to same colour column.

Let us take an example  Where  A =

Step 1 Then  on transforming 1st Column  to 1st Row  ,we have 

4             -3         9          as 1st  Row

Step 2 Then  on transforming  2nd Column  to 2nd Row  ,we have 

5             2          -2          as  2nd Row

||ly    on transforming   3rd  Column  to 3rd Row  ,we have 

7          3          8             as 3rd Row

After  taking Transpose  A' will be  

Let us take more example to find out the transpose of matrices. These are two examples , both of which are of  3×3 orders. Hence the transpose of these matrices will be  again 3×3. 


1st consider matrix G , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  -2,  -5,  4.
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  G '  will be   -5,  7 ,  3. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  G '  will be  4,  3,  8. If we write the transpose of  matrix G then we can see that there is no difference between the matrix G and the transpose of the matrix G.

Now consider matrix H , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  2,  3,  4.      
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  H '  will be   3, 5 ,  6. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  H '  will be  4,  6,  7. Again in the case of matrix H , Matrix H and its Transpose matrix   H '  are same.

Here is more interesting Example of Transpose of this Matrix,This matrix have 3 Rows and 3 columns,after taking Transpose this matrix still have 3 Rows and 3 columns,

Now Take Transpose of this Matrix 

what ?

Surprise to see that the Transpose of some of the Matrices are the Matrices Itself, i. e. if A¹= A

Such Matrices are called Symmetric  Matrices.

Again if we take the transpose of the matrix given below and take -1 common from the matrix so obtained i .e  this matrix will be equal to transpose of the negative of the transpose of the given matrix

then   A¹ =  - A ,  Such Matrices are called  Skew Matrices.

Final words

This post was regarding what is transpose of matrix, calculate  transpose , define transpose,meaning of transpose , If you learn  something from this post then share it  with your friends  and also follow me on my blog ,We  shall meet  again in next post , till then  Good Bye ..................................

Read More