Question

Consider the following functions

f(x) =x^2, g(x) = lnx, h(x) = cosx

For each of the following parts, you may use compositions, products, and sums of thefunctions above, but no others. For example, we can combine in the following waysh(g(x)) = cos(lnx), or g(x)h(x) = lnxcosx, or g(x) +h(x) = lnx+ cosx

show how derivative rules apply to the function you came up within order to produce the requested derivative.

1)A functionk(x) whose derivative is k′(x) = −tanx= -(sinx/cosx)

2) A functionj(x) whose derivative is j′(x) = x(1 + 2 lnx) =x+ 2x ln x

3) A function l(x) whose derivative is l′(x) = −2 cosxsinx

Answer #1

1.
What is the derivative of f(x) = cosx/sinx
2. What is the derivative of f(x) = cos^-1 (3x)
3. What is the second derivative of tanx/secx
4. True or False: If f'(x) = 2^x, then a possible equation for
f is f(x) = 2^x +3
5. True or False: The equation x^2 + y^2 = 100 is an implicit
curve

Having knowledge of composite functions, we know that the
mastery of some functions are the image of others, that is, a
composite function H (x) can be given by H (x) = f (g (x)). Many
functions of this type are transcendent, meaning that they have no
algebraic formulation. Given that if f (x) = sen (x), f ’(x) = cos
(x), and considering your knowledge of the chain rule for
derivation of composite functions, analyze the following
statements.
I....

Consider the following five utility functions.
G(x,y) = x2 + 3 y2
H(x,y) =ln(x) + ln(2y)
L(x,y) = x1/2 + y1/2
U(x,y) =x y
W(x,y) = (4x+2y)2
Z(x,y) = min(3x ,y)
In the case of which function or functions can the Method of
Lagrange be used to find the complete solution to the consumer's
utility maximization problem?
a.
H
b.
U
c.
G
d.
Z
e.
L
f.
W
g.
None.

Find the derivative of the following functions
(a) f(x) = ln(√x3 −2x)
(b) g(x) =√x2 + 3 x3 −5x + 1
.

13.1.7. Problem. Let f(x) = sinx and g(x) = cosx for 0 ≤ x ≤ π.
Find du(f,g) in the set of functions B([0, π]).
13.1.8. Problem. Let f(x) = 3x−3x3 and g(x) = 3x−3x2 for 0 ≤ x ≤ 2.
Find du(f,g) in the
set of
functions B([0, 2]).

(a)(3p) Discuss the continuity of the following functions:
(i)(1.5p)
f(x)= (cosx if x<0, 2 if x=0, 1-x^4 if x>0)
at x=0
(ii)(1.5p)
f(x)= 5-2x if x<-3, x^2 + 2 if x>= -3
at x=−3

#1: Consider the following.
g(x) = 9x2 − 8; h(x) =
1.1x
Find the derivative of f(x) = g(x) · h(x)
f '(x) =
#2: Consider the following.
g(x) = 2e6.5x; h(x) =
2(6.5x)
Find the derivative for f(x) = g(x) · h(x)
f '(x) =
#3: Consider the following.
g(x) = 8e−x + ln x; h(x) =
9x2.7
Find the derivative for f(x) = g(x) · h(x)
f '(x) =

Consider the functions f(x) and g(x), for which f(0)=2, g(0)=4,
f′(0)=12 and g′(0)= -2
find h'(0) for the function h(x) = f(x)/g(x)

Diﬀerentiate the following functions. Simplify your answers.
SHOW ALL STEPS.
A. h(x) = (e^3x −2)^5
B. A(x) = lnx/2x^6

Find the derivatives of each of the following functions. DO NOT
simplify your answers.
(a) f(x) = 103x (3x5+ x − 1)4
(b) g(x) = ln(x3 + x) /
x2 − 4
(c) h(x) = tan-1(xex)
(d) k(x) = sin(x)cos(x)

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