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, Product Rule, Quotient Rule, and Composite Rule

In each of the following parts, you should simplify your answers where it is appropriate to do so. (1) (a) Write down the derivative of each of the functions f(x) = e^-8x and g(x)=sin(4x) (b) Hence, by using the Product Rule, differentiate the function k(x) = e^-8xsin (4x) (2) (a) Write down the derivative of eac

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

Finding Derivatives Using Differentiation Rules

Find derivatives of the following functions using differentiation rules: 1. f(x) = 9x+5 2. f(x) = 3e^x 3. f(x) = 3x^3 + 4 4. f(x) = x^4 + 3x^2 + 4x - 31 5. f(x) = (2x + 3)e^x 6. f(x) = (2x - 1)e^(x-3) 7. f(x) = x(x-2)^2 = x(x^2 - 4x + 4) = x^3 - 4x^2 + 4x

Calculus Review: Maxwell-Boltzmann Distribution and Derivatives.

An explanation with work would be greatly appreciated. Thank you. Math Practice 1. For F (x,y) = e^(xcos(y)) find dF/dx, dF/dy, d^2F/dx^2, d^2F/dxdy 2. Normalize the Maxwell-Boltzmann distribution. More specifically, solve for N below: (See attachment for full equations.)

Evaluating Limits and Derivatives

I need some help with these three questions: A. Evaluate lim x->2 for (5x^2 + 60x + 100)/ (x^3 + 4x^2 + 4x) B. Evaluate the integral (256,1) for ((x^1/2)/4 + 8/x^8)) dx C. Evaluate the integral (6,2) for (10x-72x^-3) dx See attachment for better formula representation

Find the intervals

A) Find the intervals where g(x)=2x^3-96x^2+1440x-3676 is increasing and the intervals where it is decreasing. B) Find the intervals where h(x)=-1/2x^4+3x^3+54x^2+9x-1 is concave up and concave down. Help would be much appreciated.

Convex Subset and Lipshitz Condition

Please solve the following problem: Let E be an open and convex subset of R^n and let f in C^1(E). Show that f satisfies the Lipschitz condition on E if and only if its derivative Df is bounded on E, that is, there exists a constant M => 0 such that the norm of IIDf(x)II =< M for all x in E, where IIDf(x)II is the norm of

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.

Initial Value Problem

For every xo in R, solve the initial value problem x' = IxI , x(0) = xo and show that the solution is unique on (minus infinity, plus infinity), where f (x) = IxI is the absolute value of x. Hint: consider two cases: i) xo >= 0 and ii) xo < 0.

Finding function derivatives

Please see the attached file for the complete problem description. a. g(alpha) = 5^-alpha/2 sin 2alpha b. y = ln(t^3 +4) - l/2 arctan t/2

Derivative Problem: Bob the Formula 1 Fan

Bob is a Formula 1 fan, lucky enough to obtain tickets for the last race of the season, and the tickets are excellent, facing the long straight part of the track, where most of the overtaking occurs. At one point during the race, the race leader reached the top speed of 300km/h, and he was almost in front of Bob when it happened

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

Implicit differentiation, equation of the tangent line

Please see the attached problem for the full questions. 1) Find f'(x) and f'(c): f(x) = x^2 - 4 / x - 3, c =1 2) Find dy/dx by implicit differentiation: sinx + 2cos2y = 1 3) A) Use implicit differentiation to find an equation of the tangent line to the ellipse: x^2/2 + y^2/8 = 1 at (1,2) B) Show that the equation of

Examples of the product rule of exponents

Examples of the product rule of exponents: 2^1 * 2^2 = 2^(1+2) = 2^3. 10^2 * 10^3 = 10^(2+3) = 10^5. 10^3 * 10^(-6) = 10^(3 + (-6)) = 10^(-3) = 10^(-3). 5^4 * 5^0 = 5^(4+0) = 5 Sometimes when students see the rule they want to write it as 4^2 + 4^3 = 4^5. Can you explain why this does not work?

The product rule of exponents

Explain, in your own words, the product rule of exponents. Why does it work? Give examples. Please use a problem from page 412 #'s 27 - 38 or page 413 #'s 51 - 56, 101 as your example for this question. Please try to avoid using a problem someone has already used.

Mean Value Theorem.

Explain why not all of the hypotheses for the Mean Value Theorem hold for f(x) = 1- |x| on [-1, 2]

Tangent Line: Derivatives

See the attached file. Find the derivative of the function. f(x) = (1 - e-2x)8 f '(x) = Find the derivative of the function. f '(x) = Find the derivative of the function. f '(x) = Find an equation of the tangent line to the graph of y = e-x2 at the point (3, 1/e9). y = Find an equation of the tangent

Finding the Maximal and Minimal by Derivatives

Please help me with the following calculus problem: 1. A homeowner wants to enclose a 5,800 square feet rectangular garden by a fence in his backyard. If three sides of the fence cost $6.25 per foot and 4th side costs $10.25 per foot, find the dimensions that will minimize the cost of building the fence and the minimum cost

A formal proof of the Chain Rule

Theorem: Let X, Y be subsets of R, let x_0 belong to X be a limit point of X, and let y_0 belong to Y be a limit point of Y. Let f:X-->Y be a function such that f(x_0)=y_0, and such that f is differentiable at x_0. Suppose that g: Y-->R is a function which is differentiable at y_0. Then the function g o f: X-->R is differentiabl

Proof differentiation of functions

1) Let n be a natural number, and let f: R-->R be the function f(x) :=x^n. Show that f is differentiable on R and f '(x)=nx^n-1 for all x belonging to R (Use induction) 2) Let n be a negative integer, and let f:R - {0} -->R be the function f(x):=x^n, show that f is differentiable on R and f '(x)=nx^n-1 for all x belonging t

Using Derivatives to Find Tangent Line and Rate of Change

Please see the attached file for correctly formatted equations. (3) Find the equation of the line tangent to g(x) = -6x + 64sqrt(x) at x = 16. (4) Find the average rate of change of h(x) = -3x^4 + 14x^2 + 22x from x = -3 to x = 5.

Using Differentiation to Find Out the Minimal and Maximal

A meteorologist sketched the path of the jet stream on a map of the northern U.S. and southern Canada on which all latitudes were parallel and all longitudes were parallel and equally spaced. A computer analysis showed this path to be given by: Find the locations of the maximum and minimum latitudes of the jet stream between

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

Differentiation of transcendental functions

Solve the given problem by finding the appropriate derivative. Find the equation of the line normal to the curve of y= 2 cos (1/2)x, where x=pi. Write the expression using x as a variable. Write the exact answer in terms of pi.

interest rate derivative

The one-month and two-month interest rates are 3.8% and 4.0%, respectively. Our model of the term structure says that one month from now the one-month interest rate will be either 3.3% or 4.3%. Compute the price of an interest rate derivative that pays $1 in one month if the one-month interest rate is 4.3% and $0.5 if the one-mo

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