**2.** **5** **Derivatives** **of** **Higher**
**Orders**

**2.5.1 Definition of higher order derivatives****:** A *derivative of the second order*
or the *second derivative* of the function *y=f*(*x*)
is the derivative of its derivative, i.e.,

The second derivative may be represented by

If *x = f*(*t*) is the law of
rectilinear motion of a point, then *d*²*x*/*dt*²*
*is the acceleration of this motion.

In general, the *n-*th derivative of a
function *y = f*(*x*) is the derivative of a derivative
of order (*n - *1). We use for the notation of the *n-*th
derivative

**Example 1.** Find the second
derivative of the function

**Solution:**

**2.5.2
Leibnitz's rule**: If the
functions *u* = *j*(*x*) and *v* = *y*(*x*) have
derivatives up to the *n*-th order inclusively, then we
can use the *Leibnitz's rule* (or formula) to evaluate the
*n-*th derivative of a product of these functions:

**2.5.3 Higher-order derivatives of parametrically
represented functions****:**
If

then the derivatives y'_{x}* = dy*/*dx*,
*y*"_{xx}, ··· can successively be
calculated by means of the formulae

For a second derivative, we have the formula

**Example 2.** Find *y**"**,*
if

**Solution:** We have

A. Higher-order derivatives of explicit functions

Find the second derivatives of the functions:

**B.**** ****Higher-Order Derivatives of Parametrically
Represented Functions and implicit Functions**

Find* **dy*^{2}/*dx*^{2
}in the following problems:

**EXERCISES 699 - 711**

In the following examples, find *y''' = d*³*y**/dx*³:

**2.6 Differentials of First and Higher Orders**

**2.6.1
First-order differential****:**
The *differential (first-order) of a function*** **is
the principal part of its increment, which part is linear
relative to the increment D*x = dx* of the independent variable *x.* The
differential of a function is equal to the product of its
derivative and the differential of the independent variable

whence

If *MN* is an arc of the
graph of the function y = f(x) (Fig. 19), MT is the tangent at
M(x, y*)* and

then the increment in the ordinate of the tangent

and the segment *AN = *D*y.*

**Example 1.** Find
the increment and the differential of the function *y *= 3*x*²*
- x.*

**First solution:**

whence

**Second solution:**

**Example 2:**
Calculate *D**y* and *dy* of the function *y = *3*x*²*
-* *x* for *x*= l and D*x* = 0.01.

**Solution:**

**2.6.2 Principal properties of differentials****:**

**2.6.3. Applying differentials to approximate
calculations****:**.
If the increment D*x *of the argument *x* is small in absolute
value, then the differential *dy* and the increment Dy of the function *y
= f*(*x*) are approximately the same:

**Example 3.** By (approximately)
how much does the side of a square change if its area increases
from 9² m* to 9.1 m² ?

**Solution:** If *x* is the
area of the square and *y* is its side, then

We know that *x* = 9 and D*x* = 0.1.
The increment D*y* of the side of the square may be calculated
approximately as follows:

**2.6.4
Higher order differentials****:**
A *second-order differential* is the differential of a
first-order differential:

We define similarly the *differentials of
the third *and* higher orders*. If *y = f*(*x*)
and *x* is the independent variable, then

However, if *y = f*(*u*)*,*
where *u* = *j*(*x*), then

(Here the primes denote derivatives with
respect to *u).*

**712.** Find the increment D*y** *and the
differential *dy* of the function *u = *5*x + x*²
for *x = *2 and D*x* = 0.001.

**713.** Without calculating the
derivative, find

714. The area of a square *S* with side *x*
is given by *S = x*²*'.* Find the increment and the
differential of this function and explain the geometric
significance of the latter.

**715. **Give a geometric
interpretation of the increment and differential of the
functions:

a) the area of a circle, *S*
= *p**x*²,

b) the volume of a cube, *v =x*³*.*

**716.** Show that when D*x *®* *0*,* the increment in the function *y = *2^{x}*,*
corresponding to an increment D*x* in *x,* is for any *x* equivalent to
the 2^{x} ln 2 D*x.*

**717.** For what value of *x*
is the differential of the function *y =x*² not equivalent
to the increment in this function as D*x*®0?

**718.** Has the function *y =
|x|* a differential for *x = *0?

**719.** Using the derivative, find the
differential of the function *y= *cos* x* for *x = **p*/6* *and
D*x
= *p*/*36*.*

720. Find the differential of the function

for *x = *9 and D*x = -* 0.01.

**721.** Calculate the
differential of the function

for *x* = *p*/3 and D*x ***=*** p***/**180**.**

In the following problems, find the differentials of the given functions for arbitrary values of the argument and its increment:

**731.** Find *dy* if *x*²
*+ *2*xy - y*² *= a*²*.*

**Solution:** Taking advantage of
the invariance of the form of a differential, we obtain 2*xdx *+
2(*ydx + xdy*) = 2*ydy *=* *0, whence

In the following examples find the differentials of functions defined implicitly.

**735.** Find *dy* at the
point (1,2), if *y*³ *-* *y = *6*x*²*.*

**736.** Find the approximate
value of sin 31°.

**Solution:** Setting *x *= arc 30º
= *p */6 and D*x = *arc 1º = *p */180º, we have
sin 31º » sin 30º + *p*/180 cos 30º = 0.500+0.017Ö3/2 = 0.015.

**737.** Replacing the increment
of the function by the differential, calculate approximately:

**738.** What will be the
approximate increase in the volume of a sphere if its radius *R
= *15 cm increases by 2 mm?

**739.** Derive the approximate
formula, for |D*x*| which are small compared with *x**,*

Using it, approximate

**740.** Derive the approximate
formula

and find approximate values of

**741.** Approximate the
functions:

**742.** Approximate tan 45° 3' 20".

**743.** Find the approximate
value of arsin 0.54.

**744.** Approximate

**745.** Using Ohm's law *I *=
*E*/*R**,* show that a small change in the
current, due to a small change in the resistance, may be found
approximately by the formula

**746.** Show that, in determining
the length of the radius, a relative error of l% results in a
relative error of approximately 2% in a calculation of the area
of a circle and the surface of a sphere.

**747.** Compute *d*²*y,*
if *y* *=* cos 5*x.*

**Solution:**

**2.7.1
Rolle's theorem****:**
If a function *f*(*x*) is continuous on the interval *a
*£ *x **£** b,* has a derivative *f *'(*x*) at
every interior point of this interval and

then the argument *x* has at least one
value *x*, where a **<** *x ***<** *b*
such that

**2.7.2
Lagrange's theorem****:**
If a function *f*(*x*) is continuous on the interval *a
< x < b* and has a derivative at every interior point of
this interval, then

where a **<** *x*** < ***b.*

**2.7.3
Cauchy's theorem****:**
If the functions *f*(*x*) and *F*(*x*) are
continuous on the interval *a < x < b,* have there
derivatives which do not vanish simultaneously and *F*(*b*) ¹ *F*(*a*), then

**756.** Show that the function *f*(*x*)*
= x - x*² satisfies on the intervals -l £ *x *£ 0 and 0 £ *x* £ l Rolle's
theorem. Find the values of *x*.

**Solution:** The function *f*(*x*)
is continuous and differentiable for all values of *x* and
*f*(-l) = *f*(0) =* f*(l) = 0. Hence, Rolle's
theorem is applicable on the intervals -1 £* x *£* *0 and 0 £ *x *£ l. In order to
find *x, *we form the equation

whence

where -1 < *x*_{1} < 0
and 0 < *x*_{2} < 1.

**757.** The function

has equal values

at the end-points of the interval [0,4]. Does Rolle's theorem hold for this function on [0,4]?

**758.** Does Rolle's theorem hold
for the function

on the interval [0, *p*)?

**759.** Let

Show that the equation has three real roots.

**760.** Obviously, the equation

*f '*(*x*)* = *0 has a root *x
= *0*.* Show that this equation cannot have any other
real root.

**761.** Test whether Lagrange's
theorem holds for the function

in the interval [-2, 1) and find the
appropriate intermediate value of *x.*

**Solution:** The function is
continuous and differentiable for all values of *x* and *f
*'(x) = l - 3*x*², whence, by Lagrange's formula, we
have

Hence, 1 - 3*x*² = - 2 and *x *= ± l; the only suitable value is *x *= -1, for which -2
< x
< 1.

**762.** Test the validity of
Lagrange's theorem and find the appropriate intermediate point *x* for the
function *f*(*x*)* = x*^{4/3} in the
interval [-1, 1].

**763**. Given a segment of the
parabola *y = x*² lying between two points *A*(l, l)
and *B*(3*,* 9)*,* find a point at which the
tangent is parallel to the chord *AB.*

**764.** Using Lagrange's theorem,
prove the formula

where *x < **x **< x + h.*

**765.** a) For the functions *f*(*x*)*
= x*² *+ *2 and *F*(*x*)* = x*³* - *1,
test whether Cauchy's theorem holds on the interval [1, 2] and
find *x*;

b) Repeat for *f*(*x*)* = *sin *x* and *F*(*x*)*
= *cos *x* on the interval [0, *p*/2].

If a function *f*(*x*) is
continuous and has continuous derivatives up to the (*n -*
l)-th order inclusively in the interval 0 £ :*x*
£ *b* (or b £ *x *£* a),* and there is a finite derivative *f*^{
(n)}(x) at each interior point of the interval,
then *Taylor's formula*** **

where *x
*= *a *+ *q *(*x* - *a*)and
0 < *q *< l, holds true in the interval.

In particular, when *a = *0, one has **Maclaurin's
formula**

where *x
*= *q
**x*, 0 < *q* < l.

**766**: Expand the polynomial

in positive integral powers of *x—*
2.

**Solution:**

for *n = *4, whence

Thus,

or

**767.** Expand the function *f*(*x*)*
= e*^{x }in powers of *x + *1 to the term
containing (*x +* 1*)*³*.*

**Solution: ***f*'^{(n)}(*x*)
**= ***e*^{x} for all *n*, *f*^{
(n)}(-l)= 1/*e*, whence

where *x
*= - l + *q *(*x* + 1),
0 < *q *< l.

**768.** Expand the function *f*(*x*)*
= *ln* *(*x*) in powers of *x - *1 up to the
term with (*x -* l)².

**769.** Expand *f*(*x*)
*= *sin* x* in powers of *x* up to the term
containing *x*^{3} and to the term containing *x*^{5}*.*

**770.** Expand *f*(*x*)*
= e*^{x} in powers of *x* up to the term
containing *x*^{n-1}*.*

**771.** Show that sin (*a*
+ *h*) differs from

by not more than 1/2 *h*².

**772.** Determine the origin of
the approximate formulae:

and evaluate their errors.

**773.** Evaluate the error in the
formula

**774.** Due to its own weight, a
heavy suspended thread lies in a catenary *y = a* cosh *x*/*a*.
Show that for small |*x*| the shape of the thread is
approximately expressed by the parabola

775*. Show that for to within (*x*/*a*)²*,*
we have approximately