Derivatives and Integrals of Exp and Log Functions

1. Find the derivatives for the following functions ("^" means "to the power of", sorry I can't do double exponents on my keyboard) :

a. f(X) = 100e10X

b. f(X) = e(10X-5)

c. f(X) = e^X3

d. f(X) = 2X2e^(1- X2)

e. f(X) = 5Xe(12- 2X)

f. f(X) = 100e^(X3 + X4)

g. f(X) = e^(200X - X2 + X100)

2. Find the derivatives for the following functions:

a. f(X) = ln250X

b. f(X) = ln(20X-20)

c. f(X) = ln(1- X2)

d. f(X) = ln(5X + X-1)

e. f(X) = Xln(12- 2X)

f. f(X) = 2Xln(X3 + X4)

g. f(X) = ln(200X - X2 + X100)

3. Find the indefinite intgrals for the following functions

a. f(X) = e6X

b. f(X) = e(5X-5)

c. f(X) = 5eX

d. f(X) = 1/(1+X)

e. f(X) = 5/X

4. Find the definite intgrals for the following functions

a. f(X) = e2X over the interval [2, 4]

b. f(X) = 2eX over the interval [0, 2]

d. f(X) = 2/(2+X) over the interval [2, 5]

e. f(X) = 10/X over the interval [3, 10]

Solution Summary

Twenty-four problems involving Derivatives and Integrals of Exponential and Logarithmic Functions are solved. The response received a rating of "5" from the student who originally posted the question.

With the exponential function e^x andlogarithmic function log x how do I graphically show the effect if x is doubled?
I need to also calculated the values for e^x and e^(2x) and plotted the values of e^x and e^(2x)
then I need to also calculated the values for log x andlog 2x and plotted the values of log x andlog 2x .

Consider the formal power series f(x) = x + x^2/2 + x^3/6 + x^4/24 and g(x) = x - x*/2 + x^3/3 - x^4/4. Compute by hand the first five coefficients (i.e., up to the coefficient of x^4) of
(a) h(x) = x^f(x)
(d) k(x) = log(1 + g(x))
(c) m(x) = (h o k)(x)

1. The double integral (see attached) is determined in the domain D shown in the figure below. The domain boundaries are determined by the following functions:
1) y1(x) = x;
2) y2(x) = (9-x^2)^(1/2);
3) y3(x) = -x;
4) y4(x) = (1-x^2)^(1/2)
(a) Simplify the integrant, reduce the integral to the polar coordinates and indic

(See attached file for full problem description)
1. Calculate the values of the following integrals.
2. Write delta(x^2 - 4) in terms of a sum of ordinary delta functions.
3. Write delta(sin(x)) in terms of a sum of ordinary delta functions (an infinite number!).
4. Calculate the values of the following integrals inv

A system is composed of N one dimensional classical oscillators. Assume that the potential for the oscillators contains a small quartic "anharmonic" term
V(x) = (m*Ω^2)/2 + a*x^4
Where a*(x^4) <<< KB T and (x^4) = average value
Calculate the average energy per oscillator to the first order in a

See the attached file.
Solve the integrals of the functions:
x^2 * exp (2x)
t^2 * sin(t)
x ln(x)
exp(3z) * cos(3z)
ln(x) / x^2
t * [ln(t)]^2
x^3 * sqrt (1-x^2)
sin[ln(t)]
ln(1+x^2)
x^2 * cos (4x).