Differentiation Rules CHAPTER 3 264 d [tanh (sin x)]. dx EXAMPLE 5 Find SOLUTION Using Table 6 and the Chain Rule, we have 1 (sin x) [tanh (sin x)] dx (sin x)2 dx 1 COSX =sec x COS x= cosx 1 sin2x 3.11 EXERCISES 21. If cosh x =and x> 0, find the values of the other hyper- bolic functions at x 1-6 Find the numerical value of each expression. (b) cosh 0 1. (a) sinh 0 22. (a) Use the graphs of sinh, cosh, and tanh in Figures 1-3 to (b) tanh 1 2. (a) tanh 0 draw the graphs of csch, sech, and coth. (b) Check the graphs that you sketched in part (a) by using a graphing device to produce them. (b) cosh 5 3. (a) cosh(In 5) 4. (a) sinh 4 (b) sinh(ln 4) 23. Use the definitions of the hyperbolic functions to find each of the following limits. (a) lim tanh x 5. (a) sech 0 (b) cosh1 (b) sinh 1 6. (a) sinh 1 (b) lim tanh x (d) lim sinh x (c) lim sinh x 7-19 Prove the identity (e) lim sech x (f) lim coth x 7. sinh(=x) (This shows that sinh is an odd function.) = -sinh x (g) lim coth x (h) lim coth x 8. cosh(=x) = cosh x (This shows that cosh is an even function.) sinh x (i) lim csch x (j) lim r 0 et 9. cosh x +sinh x = e 24. Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth. 10. Cosh x sinh x e * 11. sinh(x+ y) = sinh x cosh y + cosh x sinh y 25. Give an alternative solution to Example 3 by letting y sinhx and then using Exercise 9 and Example 1(a) with x replaced by y. 12. cosh(xy) = cosh x coshy + sinh x sinhy 13. coth2x- 1 = csch?r 26. Prove Equation 4. tanh x + tanhy 14. tanh(x+y) 27. Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with x replaced by y. 1 +tanh x tanh y 15. sinh 2x2 sinh x cosh x 28. For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. (a) csch 16. cosh 2x =cosh2r +sinh'x x2 1 17. tanh(In x) (b) sech x21 (c) coth 1 + tanh x 18. 1 - tanh x 29. Prove the formulas given in Table 6 for the derivatives of the following functions. (a) cosh (d) sech e 2x 19. (cosh x + sinh x)" = cosh nx + sinh nx (n any real number) (b) tanh (e) coth (c) csch 30-45 Find the derivative. Simplify where possible. 20. If tanh x 1, find the values of the other hyperbolic func- 30. f(x)e cosh x tions at x. 31. f(x) tanh Vx 32. g(x) sinh2x = 2
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
I need help with question 21 in Section 3.11, page 264, of the James Stewart Calculus Eighth Edition textbook.
![Differentiation Rules
CHAPTER 3
264
d
[tanh (sin x)].
dx
EXAMPLE 5 Find
SOLUTION Using Table 6 and the Chain Rule, we have
1
(sin x)
[tanh (sin x)]
dx
(sin x)2 dx
1
COSX
=sec x
COS x=
cosx
1 sin2x
3.11 EXERCISES
21. If cosh x =and x> 0, find the values of the other hyper-
bolic functions at x
1-6 Find the numerical value of each expression.
(b) cosh 0
1. (a) sinh 0
22. (a) Use the graphs of sinh, cosh, and tanh in Figures 1-3 to
(b) tanh 1
2. (a) tanh 0
draw the graphs of csch, sech, and coth.
(b) Check the graphs that you sketched in part (a) by using a
graphing device to produce them.
(b) cosh 5
3. (a) cosh(In 5)
4. (a) sinh 4
(b) sinh(ln 4)
23. Use the definitions of the hyperbolic functions to find each
of the following limits.
(a) lim tanh x
5. (a) sech 0
(b) cosh1
(b) sinh 1
6. (a) sinh 1
(b) lim tanh x
(d) lim sinh x
(c) lim sinh x
7-19 Prove the identity
(e) lim sech x
(f) lim coth x
7. sinh(=x)
(This shows that sinh is an odd function.)
= -sinh x
(g) lim coth x
(h) lim coth x
8. cosh(=x) = cosh x
(This shows that cosh is an even function.)
sinh x
(i) lim csch x
(j) lim
r 0
et
9. cosh x +sinh x = e
24. Prove the formulas given in Table 1 for the derivatives of the
functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth.
10. Cosh x sinh x e *
11. sinh(x+ y) = sinh x cosh y + cosh x sinh y
25. Give an alternative solution to Example 3 by letting
y sinhx and then using Exercise 9 and Example 1(a)
with x replaced by y.
12. cosh(xy) = cosh x coshy + sinh x sinhy
13. coth2x- 1 = csch?r
26. Prove Equation 4.
tanh x + tanhy
14. tanh(x+y)
27. Prove Equation 5 using (a) the method of Example 3 and
(b) Exercise 18 with x replaced by y.
1 +tanh x tanh y
15. sinh 2x2 sinh x cosh x
28. For each of the following functions (i) give a definition like
those in (2), (ii) sketch the graph, and (iii) find a formula
similar to Equation 3.
(a) csch
16. cosh 2x =cosh2r +sinh'x
x2 1
17. tanh(In x)
(b) sech
x21
(c) coth
1 + tanh x
18.
1 - tanh x
29. Prove the formulas given in Table 6 for the derivatives of the
following functions.
(a) cosh
(d) sech
e 2x
19. (cosh x + sinh x)" = cosh nx + sinh nx
(n any real number)
(b) tanh
(e) coth
(c) csch
30-45 Find the derivative. Simplify where possible.
20. If tanh x 1, find the values of the other hyperbolic func-
30. f(x)e cosh x
tions at x.
31. f(x) tanh Vx
32. g(x) sinh2x
=
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