Share
Explore BrainMass

# Derivatives

A Derivative is a measure of how the output of a specific function, which is not limited to y or f(x), changes with respect to the input. It is commonly written in the following form:

dy/dx

where,

d also known as delta, represents the change

y is the output

x is the input

Thus for a simple function such as y=5x^3, we can differentiate it in the following way:

Y = 5x^3

(d/dx)Y = (d/dx)5x^3

dy/dx = 5(d/dx)x^3

dy/dx = 5(3x^2)

dy/dx = 15x^2

However, finding the derivatives is usually never as easy as the above example. Instead there are many different methods to find the derivative of complex functions. For example, there is the quotient rule for functions with fractions; and there is the chain rule for composition functions. Consider the following function:

y = (2x+2)/(x-5)

we can split the function into two separate functions:

f(x) = 2x+2

g(x) = x-5

The quotient rule to solve this derivative is as follows:

[(f(x)/g(x)]’ = [g(x)*f(x)’ – f(x)*g(x)’]/[g(x)^2)

dy/dx = [(x-5)(2) – (2x+2)(1)]/[(x-5)^2]

dy/dx = [2x-10-2x-2]/[(x-5)^2]

dy/dx = -12/[(x-5)^2]

From this example, it can be seen that finding the derivative is not always straightforward.  Thus, understanding the complexities and the multitude of different rules to differentiate a function is crucial for the study of calculus.

### Position of a Nonnegative Differentiable Function on a Closed Bounded Interval

Let f(x), g(x) be functions defined on a closed bounded interval [a, b] such that the following conditions hold: g is differentiable on [a, b]. There are positive constants a, b such that g(x) = a*f(x) - b*(dg/dx). f(x) > 0 for all x in [a, b] g(x) >= 0 for all x in [a, b] g(a) > 0 -----------------------------

### Derivatives / Chain Rule / Taylor Polynomials

If I have the following function f(x,y) = x^3 + x^2y − 16y There are 6 things I would like to know how to do: I would like to know how to find the ﬁrst-order and second-order partial derivatives of f? If the value of f when x = 3 and y = −2 is 41 but I was to know only that the value of x lies between 2.98 and 3.0

### Evaluating Stationary Points

Using the function f(x)=2x^3+3x^2-36x+7 a) Find the stationary points of this function. b) i) Applying First Derivative Test, classify left hand stationary point in part a. ii) Applying First Derivative Test, classify right hand stationary point in part a. c) Find the y coordinates of each stationary point on the graph of th

### Returns to Scale and Marginal Product of Capital and Labor

See the attached file. We give step by step solution to the following question. Let f(K,L) be a production function with constants returns to scale, where K denotes capital and L denotes labor. (a) Show that if we scale both input factors up or down by t>0, the marginal products of labor and capital remain the same.

### Relative Extrema, Production Level

See the attached file for the problem. 1. Find all relative extrema. Use the Second Derivative Test if applicable. g(x) = x^2(6-x)^3 2. A manufacturer has determined that the total cost C of operating a factory is C = 0.5x^2 + 15x + 5000, where x is the number of units produced. At what level of production will the avera

### Derivatives

The percent of mothers who work outside the home and have children younger than age 6 yr is approximated by the following function where t is measured in years, with t = 0 corresponding to the beginning of 1980. P(t) = 34.57(t + 3)0.205 (0 t 21) Compute the following value. Round your answer to 4 decimal places. P''(1

### Find the derivative f 'of f and tangent line

Let f(x) = x2 + 4x. (a) Find the derivative f 'of f. (b) Find the point on the graph of f where the tangent line to the curve is horizontal. Hint: Find the value of x for which f '(x) = 0. (c) Sketch the graph of f and the tangent line to the curve at the point found in part (b). Find the slope m of the tangent line to

### Calculus with a lot of examples

Let f be defined as follows. f(x)=x^2-3x (a) Find the average rate of change of y with respect to x in the following intervals. from x = 6 to x = 7 from x = 6 to x = 6.5 from x = 6 to x = 6.1 (b) Find the (instantaneous) rate of change of y at x = 6. The demand for Sportsman 5 X 7 tents is

### Derivation of thermodynamic identities

I need the solution to #10 on the attached file. beta=(1/V)(dV/dT)_p=coefficient of expansion k=-(1/V)(dV/dp)_T= compressibility gamma=Cp/Cv Cv and Cp are just heat capacities.

### Differentiation and Limits

See the attached file. 1. Differentiate a. Y = 3x + PI^3 b. Y = 1 / (x-3)^3 c. y = (x^4 - x)^3 (3x + 2)^4 d. Y = (1 + x - x^3)^4 2. Compute the following limits. a. lim(x??)?[(x-2)/(x^2+2)] b. lim(x??)?[(3x^5- 6x^4+ 2x-6)/(7x^5- 2x^2+ 10,000)] 3. Use limits to compute f"(3) where f (x) = x^

### Find the equilibrium price and quantity derivative

Please help with the following problems. 1. You are in the market for oranges. The supply equation (in millions) for oranges is : S(P)= .3p^2 +11P - 40 The demand equation is D(P) = .7p^2 +P - 1 a. How many oranges are demanded at a price of \$11.50? b. Find the equilibrium price and quantity. 2. F(X)= x^2 +2, G(X)=

### Slope of the Tangent to a Curve

Determine the slope of the tangent to each of the following curves: 1) y=2/x^2 2) y=8x-2x^2 3) y=x^4-2/x.

### Conic Section is explored.

Equation Point 4x^2-24x-25y^2+250y-489=0 (27⁄4,5⁄2) ( 1) identify the conic section represented by the equation. ( 2) write the equation of the conic section in standard form. ( 3) identify all relevant key elements of

### Optical Illusions - Finding the dy/dx for each

There are four optical illusions that are placed over a graph. In each graph depicted, an optical illusion is created by having lines intersect a family of curves. In each case, the lines appear to be curved. Please find the value of dy/dx for the given values of x and y and please show work involved so I can learn from it.

### Analyzing graphs and finding derivatives

20. The graph below displays growth of a town's population y = P(t) over the next 3 years, where t is time in months. a. Estimate how fast the population is increasing 5 months, and 20 months from now. b. Graph y = P'(t). In graphs of questions 2, 4 and 6, determine which is the f(x) function and which is the derivative?

### Mossaic tiles question

Gilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. While at these digs, they learned how to make ceramic tiles from local artisans. After college they made use of their college experiences to start a tile manufacturing firm called Mossaic Tiles, Ltd.

### Problems Involving Differentiation of Algebraic Functions

THE DIFFERENTIATION OF ALGEBRAIC FUNCTIONS Use implicit differentiation to find the dy/dx: 3xy + x2y2 = 1 xy1/2 - 2x + y = 8 (x + y)3 = x3 + y3 3y2 + 5x2 -2x = 5 y4 = x3y2 + x2y3 - 3 Find the indicated higher order derivative of the following functions: 1. Find the 2nd derivative of f(x) = 7x^5 - 4

### Limits and Derivatives

1. Find the following limits. a) Limit as x approaches 3 of: (x^2-x-6)/(x-3) b) Limit as theta approaches 0 of: (sin(3 theta))/(sin(5 theta)) 2. Using definition of derivative find the derivative of: a) f(x)= 4x^2-6x-5 b) f(x)= (1-x)/(x+1) 3. Differentiate and simplify: a) y=cos

### 4 small questions

Please show all work 1)It has been conjectured that a fish swimming a distance of L ft at a speed of V ft/sec relative to the water and against a current flowing at the rate of U ft/sec (u<v) expends a total energy given by: E(v)= aLv^3/v-u where E is measured in foot-pounds (ft-lb) and a is a constant. Find the speed V

### Find the derivative in Calculus

Find the derivative; G(v)= (v^3-1)/(v^3+1) Find the limit; lim(sin3x)/(sin5x) x->0 Find the derivative; R(w)= (cosw)/(1-sinw) H(o)=(1+seco)/(1-seco) Find the derivative; F(x)= cos(3x^2)+{cos^2}3x N(x)=(sin5x-cos5x)^5 "Assume that the equation determines a differentiable function f such that y=f(x),

### Differentiation

1. Find the rate of change dy/dx where x = x0 (Compute the derivative of the function from the definition only, using limits. Show all steps.) y = 1/(2-x), x0 = -3 2. Differentiate the function. Simplify your answer. f(x) = (1/4)x^8 - (1/2)x^6 - x +2 3. Find dy/dx by implicit differentiation. y^2 +3xy -4x^2 =

### Derivatives, Tangents and Rate of Change

All solutions must be detailed and the final answers simplified. Show all work! 1. Differentiate the given function. Simplify your answer. 2. Differentiate the given function. Simplify your answer. 3. Differentiate the given function. Simplify your answer. 4. Find the equation of the line that is ta

### Profit Function and Maximum Profit

A manufacturer finds that the total profit from producing and selling Q units of a product is given by the profit function: Total Profit = f(Q) = - 460 + 100Q - Q^2 1. Compute the value of the function at Q=10 Total Profit = f(10)= - 460 + 100(10) - 10^2 Total Profit = f(10)= - 460

### Vectors, Planes and Partial Derivatives

1. Let a = 2i + 3j and b = -9i + 6j. Find c = a - b. A) c = -3j B) c = 9i C) c = 11i + 9j D) c = 11i - 3j 2. Let a = 2i + 3j and b = -9i + 6j. Find d = a ? b. A) 36 B) -36 C) 0 D) -18i2 + 18j2 3. Find the intersection of L1: x - 2 = ½(y + 1) = 1/3(z - 3), L2: 1/3(x - 5) = ½(y - 1) = z - 4, if they

### First and Second Derivatives and Minimizing

A company holds spare parts for its car maintenance service. There is a steady demand for these parts. If the company orders large numbers once a year, then they have to pay considerable warehouse costs to stock them. If they order small numbers very frequently then they have to pay considerable admin costs for processing all th

### Calculus Problems : Continuity, Limits, Derivatives, Inequalities, Quadratic Equations and Parabolas and Maximum Height

1. Solve log&#8326;x-3=0 2. A business owner is comparing the costs of purchasing inventory and the profit from the sale of the product. The relationship proves to be linear. Which type of variation will describe the data? Direct as nth power joint regress Inverse direct 3. Solve x²-25<0 4. What is the

### Derivatives

See attachment for equations 1) determine the interval(s) where the function is increasing and the interval(s) where it is decreasing. 2) determine the interval(s) where the function is increasing and the interval(s) where it is decreasing. 3) Find the relative maxima and relative minima, if any, of the following functi

### Scheduling

See attached for proper formatting Rank the Critical ratio sequencing rules on the three evaluation criteria of average flow time, average number of jobs in the system, and average job lateness for the information below. (Do not evaluate shortest processing time or first-come first-served). A production planner must deci

### Applications of the Derivative

Please see the attached file for the fully formatted problems. For the function y= sq root (x2 - 9) (i) Find the slope of the tangent line to the function at the point (5, 4). (ii) Find the equation of the normal line at the point (5, 4). 2. A ladder 10 m long rests on horizontal ground and leans against a vertical wal

### Market Share Analysis

This table presents units sold and market share data for the Personal Computers Industry (first figure for each company is units, second is revenue). E-Top--10,000--7,500,000 CompEZ--15,000--12,000,000 BEST--2,000--1,700,000 Moonwalk--8,000--8,000,000 CompBrain--8,000--12,000,000 Market Total--43,000 units--41,200,000