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.

... quantities are functions of (derivatives of) logarithms ... We can approximately calculate the integral as follows ... you diﬀerentiate and set the derivative to zero ...

... proportional to rn−1 . The Gaussian integral is √ easily ... be ignored as they involve higher derivatives of log... the integrand and set the derivative equal to ...

... At the minimum of the potential the derivative of u must be ... The other term is exactly x_0 times the integral in the ...exp(-beta u(x_0))*exp(-beta u''(x_0)/2 y^2 ...

... (The summation can be replaced by an integration). ... of the integrand and equate the derivative wrt E1 ... have to do is to equate the logarithmic derivatives of the ...

... where the momentum integral is over a shell in momentum space that corresponds to a certain ... (1). To calculate the derivative, it is ... N(z) = N(0) Exp(-mgz/kT). ...

... For example, if the quantity is described by exponential function exp (t /T ... We define the integration factor ... The left hand side of (1.4) is a derivative of a ...

... 83, the derivative of N wrt mu will be proportional to N ...function is a trivial exercise in computing Gaussian integrals, but it ... 2 pi mk T/h^2)^(3/2) Exp[mu/(k T ...

... So you can readily evaluate the integral over theta, because the derivative of the exponent is ... Z_N = Integral over N magnetic moments of Exp[ beta (mu1 ...

... from zero to infinity of dp Omega |p|^(n-1)/h^n exp[-beta c ... Substitute x = beta c |p| in this integral to obtain. ... The average energy is given by the derivative: ...