2.7 HIGHER DERIVATIVES

DEFINITION

          The second derivative of a real function f is the derivative of the derivative of f, and is denoted by f ''. The third derivative of f is the derivative of the second derivative, and is denoted by f '', or f(3). In general, the nth derivative of f is denoted by f(n).

 

If y depends on x, y =f(x), then the second differential of y is defined to be

                           d²y = f ′′(x) dx².

In general the nth differential of y is defined by

                           dny = f(n) (x)dx n.

Here dx² means (dx)²and dxn means (dx) n.

 

We thus have the alternative notations

                 

 

 

 

For the second and nth derivatives. The notation

                             y''= f ''(x),    y(n) = f (n)(x),

is also used.

 

The definition of the second differential can be remembered in the following way. By definition,

                             dy= f '(x) dx.

Now hold dxconstant and formally apply the Constant Rule for differentiation, obtaining

                            d(dy) = f ''(x)dx dx,

or                        d²y= f ''(x)dx².

(This is not a correct use of the Constant Rule because the rule applies to a real constant c, and dx is not a real number. It is only a mnemonic device to remember the definition of d²y, not a proof.)

 

The third and higher differentials can be motivated in the same way. If we hold dxconstant and formally use the Constant Rule again and again, we obtain

 

 

 

 

 

and so on.

 

The accelerationof a moving particle is defined to be the derivative of the velocity with respect to time,

                   a = dv/dt.

 

Thus the velocity is the first derivative of the distance and the acceleration is the second derivative of the distance. If s is distance, we have

 

 

 

EXAMPLE  1  A ball thrown up with initial velocity b moves according to the equation

                                y = bt - 16t ²

 

with y in feet, t in seconds. Then the velocity is

                      v= b -32 t ft/sec,

and the acceleration (due to gravity) is a constant,

                      a= -32 ft/sec².

 

EXAMPLE  2  Find the second derivative of y=sin(2θ).

     First derivative  Put u=2θ. Then

                    

            

By the Chain Rule,

 

 

 

 

 

 

 Second derivative  Let v=2cos (2θ). We must find dv/dθ.Put u= 2 θ. Then

           

                          v=2 cos u,          

 Using the Chain Rule again,

                          

 

 The simplifies to

 

                       

 

EXAMPLE  3 A particle moves so that at time t it has gone a distance s along a straight line, its velocity is v, and its acceleration is a. Show that

             

 

 

By definition we have 

     

 

So by the Chain Rule,

            

 

  

 EXAMPLE  4  If a polynomial of degree nis repeatedly differentiated, the kth derivative will be a    

          polynomial of degree n -k for k ≤ n, and the (n+1)st derivative will be zero. For example,

 

                           

 

 

 

 

 

 

 

 

 

Geometrically, the second derivative f ′′(x)is the slope of the curve y′= f ′ (x)and is also the rate of change of the slope of the curve y=f(x).

 

PROBLEMS  FOR  SECTION 2.7

1    y=1/x                             2   y=x5                   3   y=__________

4    f(x)=3x -2                          5   f(x)= x1/2 + x -1/2         6   f(t)= t 3 - 4t 2

7    f(t) =_____                         8   y=(3t - 1)10             9   y=sin x

10   y= cos x                           11   y=A sin (Bx)

12   y= A cos (Bx)                      13  y=e ax

14   y=e -ax                            15 y=1n x

16  y= x 1n x                          17______________

18   y=_________    19_____________    20______________

21   z=________     22 _____________    23_______________

24   Find the third derivative of y= x² - 2/x.

25   A particle moves according to the equation s= 1-1/t², t >0. Find its acceleration.

26   An object moves in such a way that when it has moved a distance sits velocity is v=____.

     Find its acceleration. (Use Example 3.)

27   Suppose udepends on x and d²u/dx²exists. If y = 3u, find d²y/dx²

28   If d²u/dx²and d² v/dx²exist and y=u+v, find d²x/dx².

29   if d²u/dx²exists and y=u², find d²y/dx².

30   if d²u/dx²and d²v/dx²exist and y=uv, find d²y/dx².

31   Let y=ax2 + bx +c be a polynomial of degree two. Show that dy/dxis a linear function and d²y/dx²is

     a constant function.

□32 Prove that the nth derivative of a polynomial of degree n is constant. (Use the fact that the derivative

     of a polynomial of degree kis a polynomial of degree k-1.)