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# Derivatives

### 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

### Quotient and composite rule problem.

Please see attached problem using quotient and composite rule.

### Graph the antiderivative and the derivative of f(x).

Can you help to graph and find the antiderivative and the derivative. (See attached file for full problem description)

### Derivative of Integrals

Please see the attached file for the fully formatted problems.

### Integration: Finding a Function from a Double Derivative

Find a function f(x) such that f''(x) = x^2 + cos x and f(0) = 1 and f'(0) = 0. --- Please see the attached file for the fully formatted problems.

### 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?

### Derivatives and Limits (5 Problems)

Please see the attached file for the completely formatted problems.

### Derivatives of Trigonometric functions : Horizontal Tangent Line

Let f(x) = e sin x - cos x How are all the values of x E [-2pi, 2pi] found, such that a tangent line to the curve is horizontal and what are they? Please see the attached file for the fully formatted problems.

### 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

### Derivative of Inverse Trigonometric functions

Use inverse functions and implicit differentiation to prove that: d (tan^-1x) = 1___ dx 1+ x^2

### Product Rule for Simplification of Radicals

Use the Product rule for radicals to simplify each expression. Please see the attached file for the fully formatted problems.

### Related Rates: Rate of Change of Distance Between Objects

At noon, ship A is 100 km west of ship B. A is sailing south at 35 km/h and B is sailing north at 25 km/h. How fast is the distance between the ships changing at 16:00 h?

### 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.

### Definition of the Derivative and Limits: Slope of Tangent Line

Use the definition of the derivative to find the slope of the tangent line to the graph of f(x) = 1/(3x -2) at x = 1. See the attached file.

### Word Problem - Optimization - Applications of Derivatives: Find two numbers whose sum is 10 and product is as large as possible.

Find two numbers whose sum is 10 and product is as large as possible.

### 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.

### 14 Derivative Problems : Product Rule, Quotient Rule, Chain Rule, First and Second Derivative and Finding Maximum or Minimum

Rules and Applications of the Derivative -------------------------------------------------------------------------------- 1. Use the Product Rule to find the derivatives of the following functions: a. f(X) = (1- X^2)*(1+100X) b. f(X) = (5X + X^-1)*(3X + X^2) c. f(X) = (X^.5)*(1-X) d. f(X) = (X^3 + X^4)*(30

### 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

### Derivatives, Revenue Function, Maximizing Profit - Lemonade Stand

Data: regression equation: y= -100x + 250 regression coefficient: r= -1 X Y Predicted value 0.25 225 225 0.5 200 200 0.75 175 175 1 150 150 1.25 125 125 1.5 100 100 1.75 75 75 2 50 50 2.25 25 25 2.5

### 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 Function Provided

Find dy/dx y = 2x^3+6x^2+6x+4

### 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