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 d²x/dt² 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:

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

EXERCISES 667 - 691

Find the second derivatives of the functions:

Hints and Answers 667 - 691

EXERCISES 692 - 711

Find dy2/dx2 in the following problems:

Hints and Answers 692 - 698

EXERCISES 699 - 711

In the following examples, find y''' = d³y/dx³:

Hints and Answers 699 - 711

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 Dx = 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 = Dy.

Example 1. Find the increment and the differential of the function y = 3x² - x.

First solution:

whence

Second solution:

Example 2: Calculate Dy and dy of the function y = 3x² - x for x= l and Dx = 0.01.

Solution:

2.6.3. Applying differentials to approximate calculations:. If the increment Dx 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 Dx = 0.1. The increment Dy 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).

EXERCISES 712 - 755

712. Find the increment Dy and the differential dy of the function u = 5x + x² for x = 2 and Dx = 0.001.

713. Without calculating the derivative, find

714. The area of a square S with side x is given by S = x²'. 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 = px²,
b) the volume of a cube, v =x³.

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

717. For what value of x is the differential of the function y =x² not equivalent to the increment in this function as Dx®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 Dx = p/36.

720. Find the differential of the function

for x = 9 and Dx = - 0.01.

721. Calculate the differential of the function

for x = p/3 and Dx = 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 x² + 2xy - y² = a².

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

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

735. Find dy at the point (1,2), if y³ - y = 6x².

736. Find the approximate value of sin 31°.

Solution: Setting x = arc 30º = p /6 and Dx = 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 |Dx| 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 d²y, if y = cos 5x.

Solution:

Hints and Answers 712 - 755

2.7 Mean value theorems

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

EXERCISES 756 - 765

756. Show that the function f(x) = x - x² 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 < x1 < 0 and 0 < x2 < 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 - 3x², whence, by Lagrange's formula, we have

Hence, 1 - 3x² = - 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) = x4/3 in the interval [-1, 1].

763. Given a segment of the parabola y = x² 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) = x² + 2 and F(x) = x³ - 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].

Hints and Answers 757 - 765

2.8. Taylor's Formula

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.

EXERCISES 766 - 775

766: Expand the polynomial

in positive integral powers of x— 2.

Solution:

for n = 4, whence

Thus,

or

767. Expand the function f(x) = ex in powers of x + 1 to the term containing (x + 1)³.

Solution: f'(n)(x) = ex 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 x3 and to the term containing x5.

770. Expand f(x) = ex in powers of x up to the term containing xn-1.

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

by not more than 1/2 h².

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, we have approximately

Hints and Answers 766 - 775