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Differentiating ln x using the Quotient Rule

Question 1 says to differentiate using the quotient rule f(x) = 1 + 2x/1-2x where x < 1/2. My answer is -8xsquared/1 - 2x squared.(at x = 1/4) Question 2 says rewrite the expression of f(x) = ln (1 + 2x/1 - 2x) (-1/2 <x<1/2) by applying a rule of logarithm and then differentiate. So rewriting f (x) = ln (1 + 2x) - (1 - 2x) =

Non Linear PDE Mathematical Symbols

I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of u with respect to x. Moreover, I will further simplify things by letting p=u*x and q=u*y. Also, ^ denotes a power (for example, x^2 means x squared) and / denotes division. This is the problem: T

Applications of Derivatives: Velocity of a Particle

Let: v(t) = { 2t 0< t < 5 {10 5< t < 10 be the velocity of a particle given in meters per second. Find the distance traveled by the particle from t = 0 to t = 10 seconds. --- Please see the attached file for the fully formatted problems.

First and Second Derivatives : Using Implicit Differentiation

X^2y^2-2x=3 I'm trying to verify my answer for the first derivative, and see if I got the second one right as well. For the first derivative I got (1-xy^2)/(x^2y) I think I'm having a problem with the 2nd derivative because I got x^2-2x^3y^2+x^3y^4 It doesn't look right to me,

Derivative: Maximizing Functions and Finding a Vertex

1. Express the function in the form f(x) = a(x-h)2 + K and indicate the vertex. a. f(x) = -x^2 + 13x - 8 b. f(x) = 4x^2 - 8x + 1 2. An object is thrown upward from the top of a 160-foot (Ho) building with an initial velocity (Vo) of 48 feet per second. How long after the object is projected upward will it strike the ground?

Rates of Change : Derivatives

A tank holding 1000 gallons is being drained. The volume in the tank is given by: V (t) = 1000 (1- _t_)^2 for 0 < t < 40 40 where t is given in minutes. Find the rate at which water is draining from the tank. When is the tank draining fastest? Please see the attached file for th

Find the Derivatives (3 Problems)

Find the derivative (with respect to x) for each of the following. Do not simplify. (1) x2y3 = sqrt(2x 5) + sin(8y + 3) (2) f(x) =((8x + 3)/2x^2 3)^5 (3) y = 4th root of ((1 3x)^4 + x^4). See the attached file.

Solve the Derivative Problem

See the attached files. C(q) = 0.000002q^3 - o.o117q^2 + 84.446q + 23879 R(q) = -0.00003 * q^3 +0.0495q^2 + 118.02q P(q) = -0.000032q^3 + 0.0612q^2 + 33.554q - 23879 Use the Cost, Revenue, and Profit functions to find. a) C`(q) b) R`(q) c) P`(q) Do these equations predict the quantity needed to maximize profit, and th

Modelling the volume of a container/differentiation.

Question 1 A bucket-shaped container has a circular base of radius 10 cm, and its slant height is 30 cm. the radius of the open circular top of the container is 10x cm. the curved surface of the container is modeled by part of a cone, as shown below. Please see attached.

Differentiation Expression in Steps

Please could you solve this question clearly showing every stage used in getting to the answer. Differentiate the following expression with respect to θ... please see attached.

Derivatives, Second Derivatives and Profit Function - Lemonade Stand

A. Write a function for your profits for each price you charge. This is done by multiplying (P-.5) times your function (y= -100x + 250). I.e. if your function is Cups Sold = 1000 - 100P, your profit function would be (P - .5)*(1000 - 100P). B. Calculate the first derivative of your profit function, and create another table

Definition of a Limit and Derivative, Product Rule, Tangent Line

1. Give the definition of limit in three forms: &#949;?&#948; , graphical, and in your own words. 2. Define the derivative. List what you consider to be the five most useful rules concerning derivatives. 3. Give an argument for the product rule. 4. What is the tangent line approximation to a function? 5. What is the Taylor p

Rate of Change, Derivatives & Product and Quotient Rule

See the attached file. 71. The local game commission decides to stock a lake with bass. To do this 200 bass are introduced into the lake. The population of the bass is approximated by P(t) = 20 (10 + 7t)/(1 + 0.02 t) where t is time in months. Compute P(t) and P'(t) and interpret each. 57. The monthly sales of a new compute

Find the Function f(x)

The second derivative of a function is given as: f"(x) = 12x-1 At the point (-2,7) the tangent to the function is given by: y=kx-3 Find the function f(x).

Differentiation of Standard Functions

Differentiate each of the following functions with respect to the independent variable, use the above worked examples as an example: Please see the attached file for the fully formatted problems.

Derivative Problem Functions

Find the first derivative 1.) f(x)=e^-1/x^2 2.) f(x)= (x^2+1)e^4x 3.) y=xe^x- 4e^-x Find the second derivtive 4.)f(x)=2e^3x+3e^-2x