UNIT- 2

Differential Calculus-I

Question Bank

Question-1: Find nth derivative of log(ax + b)

Sol.

Suppose y = log (ax + b)

Differentiate with respect to x successively, we get

(-2)

.

.

.

For n times differentiation, we get-

(-2)…………….(-n + 1)

      =  …………….(n - 1)

     = (n - 1)

So we can say that its n’th derivative will be

Question-2: Find the derivative of the following function-

Sol.  Partial fraction of the function y after splitting-

        Suppose x – 1 = z, then

           =

          =

      =

Here we can find its n’th derivative-

Question-3: Find derivative of the given function:

                                y =

Sol.   We are given-

                          y =

       factorize the denominator-

                     y =

                        =

derivative will be-

Which is the derivative of the given function.

Question-4: If   y  , then prove that-

Sol. Here it is given that-

On differentiating-

Or

                  = ny.2x

Differentiate again with respect to x, we get-

Or

   …………………. (1)

Differentiate each term of (1) by using Leibnitz’s theorem, we get-

Therefore we get-

                                                                                                              Hence proved.

Question-5: If , then prove that-

Sol. Here we have-

Or

Or

                 y = b cos[ n log(x/n)]              

On differentiating, we get-

Which becomes-

Differentiate again both sides with respect to x, we get-

It becomes-

                                               ……………….. (1)

Differentiate each term n times with respect to x, we get-

Which is-

                                         hence proved

Question-6: Calculate    and   for the following function

                           f(x , y) = 3x³-5y²+2xy-8x+4y-20

Sol. To calculate   treat the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,

          = [3x³-5y²+2xy-8x+4y-20]

              = 3x³] - 5y²] + [2xy] -8x] +4y] - 20]

              = 9x² - 0 + 2y – 8 + 0 – 0

              = 9x² + 2y – 8

Similarly partial derivative of f(x,y) with respect to y is:

                      = [3x³-5y²+2xy-8x+4y-20]

                           = 3x³] - 5y²] + [2xy] -8x] +4y] - 20]

                           = 0 – 10y + 2x – 0 + 4 – 0

                           = 2x – 10y +4.

Question-7: Calculate    and   for the following function

                               f( x, y) = sin(y²x + 5x – 8)

Sol. To calculate   treat the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,

                       [sin(y²x + 5x – 8)]

                           = cos(y²x + 5x – 8)(y²x + 5x – 8)

                           = (y² + 50) cos(y²x + 5x – 8)

Similarly partial derivative of f(x,y) with respect to y is,

                      [sin(y²x + 5x – 8)]

                           = cos(y²x + 5x – 8)(y²x + 5x – 8)

                           = 2xy cos(y²x + 5x – 8)

Question-8: if , then show that-

Sol. Here we have,

u =     …………………..(1)

Now partially differentiate eq.(1) w.r to x and y , we get

                                               =

Or

                                                          ………………..(2)

And now,

                                                =

                                              ………………….(3)

Adding eq. (1) and (3) , we get

                                   =  0

Hence proved.

Question-9: If u = x²(y-x) + y²(x-y), then show that    -2 (x – y)²

Solution - here,  u = x²(y-x) + y²(x-y)

                                u = x²y - x³ + xy² - y³,

         now differentiate u partially with respect to x and y respectively,

                                         =  2xy – 3x² + y²        --------- (1)

                                       = x² + 2xy – 3y²          ---------- (2)

Now adding equation (1) and (2), we get

                                 = -2x² - 2y² + 4xy

                                               = -2 (x² + y² - 2xy)

                                               = -2 (x – y)²   

Question-10: If then prove that

Sol. Here, given-

Here u is not a homogeneous function but z =

  cos u, then

Now z is a homogeneous function in x, y of degree 1 / 2.

Now by Euler’s theorem-

Question-11: If then show that-

Sol. Here we have-

u is not homogenous function here but,

Here z is a function of degree 3.

By using Euler’s deduction formula, we get-

So that

Question-12: If u = u  then show that

Sol. Here it is given that,

                                     u = u  = u(r , s)

Where r =

Which gives-

                        and           …………………. (1)

Now, we know that-

                                      =

                                     =

Or

                                        ……………….. (2)

Similarly-

Or           

And-

Adding the results, we get

Question-13: Find the horizontal asymptote of the function-

Sol. Find the limits-

And

Hence the horizontal asymptotes are y = -1 and y = 5.

Figure will be as follows of the asymptotes of the given function-

Question-14: Find the slant asymptotes of 

Sol. We have,

Which can be written as-

The value of ‘m’ will be-

                              =     = 

Now we can find ‘c’ as –

                     c = = =

                                                                                            = 0

Hence the slant asymptotes are

Question-15: Trace the curve y = 1/x².

Sol.          As we can see y- coordinates of the curve can not be negative. So the                                              curve must be above x-axis. The curve is also symmetric about y-axis so we can draw the graph only in single side.

Here, we will find the first and second derivatives-

So, dy/dx= -2/x³ and  d²y/dx² = 6/ x⁴    ,  here  dy/dx <0 for all x>0 so we can say that the function is non- increasing so the graph falls as we increase x.

Also second derivative is also non zero so there are no point of inflection.

Here the curve is x²y=1 (rewritten), here both the axes are asymptotes of the curve.

Here is the figure of the curve-:

Question-16: Trace the curve  .

Sol. Symmetry about initial line- Put r = 0,

Hence the straight line are the tangents to the curve at the pole.

Values of ‘r’ as changes from 0 to π-

Table for the values of ‘r’-

	0	0
		Imaginary
	0	0

Here according to the table the curve does not exist for the value lying between  to

The figure will be as follows of the curve-