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# Derivatives

### Find dy/dx by differentiating implicity.

Find dy/dx by differentiating implicitly. When applicable, express the result in terms of x and y. 4(x^2 +2)^3 +(y^2+2)^2 = 19

### 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.

### Writting a Curve Graph of an Equation

Sketch a graph of an equation f(x) = (x)/(x^2) + 4) using (meaning work out each and show me your work to getting the answers) intercepts, extrema, asymptotes, symmetry, and also tell if it's concave up or down and the inflection points. Please do the first and second differentiation and tell me how you did it plus show me the

### 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^

### Derivative

The data in the table below define y as a function of x. x 0 .5 1 1.5 2 2.5 y -1.4 1.025 4.6 9.325 15.2 22.225 A. Estimate the derivative at each of the following points using the average of the left and right difference quotients. x1 0.5 1 1.5 2 slope 3.575/.5 4.725/.5 5.875/.5 7.025/

### differentiable functions.

Let g1(x), ... , gn(x) be differentiable functions. If f(x) = g1(x)...gn(x), prove that its derivative is f'(x) = SUMMATION[i=1,n] (f(x) gi'(x)/gi(x)) . Please show complete solution.

### substitution method and the quotient rule

Using the appropriate substitution method and the quotient rule find the integral of: Integral: 1+tanx/1-tanx dx Note: you may use the relationship that: tan(a+b) = tan(a) +tan(b)/1 -tan(a)tan(b) and tan(Pi/4) = 1

### Derivatives: The Chain Rule

** Please see the attached file for the complete problem description ** Find a derivative y'(x) for each of these expressions. You are advised to carefully choose a method appropriate to each expression. (please see the attached file)

### Definition of the Derivative

The derivative of a continuous function at x is the slope of the tangent line to the curve at x. The attached pdf file develops the idea of a derivative first using slopes of secant lines and then introducing and explaining the difference quotient in detail. An example and an explanation are provided for using the limit of the

### The n-th derivative of the product of two functions example

A formula is derived for the n-th derivative of a function that is a product of two other functions, f(x)=u(x).v(x) This formula is used to write down the n-th derivative of f(x) = e^x/(1 &#8722; x). The proof of the result used is by mathematical induction on 'n'. The solution is typed neatly and comes as a pdf file.

### MacLaurin Expansion of a Function

A detailed description of how to find the first three nonzero terms of the Maclaurin expansion of the function f(x)=sin2x is given.

### Calculating the splitting field greatest common divisor

Assume F has characteristic 0 and K is a splitting field of f (x) in F[x]. If d(x) is the greatest common divisor of f(x) and f ' (x) and h(x) = f(x) / d(x) in F[x], prove that a) f (x) and h(x) have the same roots in K b) h (x) is separable

### Midpoint Rule of Rectangles

Using rectangles whose height is given by the value of the function at the midpoint of the rectangles base (the midpoint rule); estimate the area under the graphs of the following functions, using first two and then four rectangles. f(x)=8-x^2 between x=-4 and x=4.

### signed curvature

Find an equation of a parabola that has curvature 4 at the origin.

### Percentage of rates and frequency of rates

When the means is 70 and the standard deviation is 15 what is the percentage of rates less than 70, what is the relative frequency of rates less than 40, What is the relative frequency of rates less than 100, What is the percentage of rates greater than 85 and what is the percentage of rates between 55 and 85?

### Product. Chain, and Quotient Rules .

Find the derivatives of the following functions, and simplify into the forms specified. (a) g(x)=x^(12) (9 ln x + 15) The derivative of g(x) can be expressed in the form g'(x)=x^(11) h(x), where h(x) is some function of x. Find h(x). h(x)=_______ (b) y= ln (3e^(9x) + x^6) dy/dx=______ (c) f(x) = (6 ln x)/5 ln x

### Differentiable Variables

Let f(x)={(x^3)cos(1/x) if xâ? 0, 0 if x=0, and g(x)={(1/x)sin(x) if xâ? 0, 0 if x=0. a) Using the definition of the derivative show that f is differentiable at 0 and determine f '(0). b) Is g differentiable at 0? Justify your answer. c) Show that f ' ^(x) and g ' ^(x) exist for xâ? 0 and determine their value

### 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.

### Maximum value of the directional derivative

Let f(x,y) = x^2 - y^3 a) Find the gradient of f(x,y) at (3,2). b) compute the directional derivative (triangle u) f(3,2) where u is the unit vector in the direction of v = <4,3> c) Find the maximum value of the directional derivative of f(x,y) at (3,2) and the direction in which it occurs.

### Differentiable Function and Expression of Equation

Suppose f is a differentiable function, such that (1) f(x+y)=f(x)+f(y)+2xy for all real numbers x and y and (2) lim_h-->0 f(h)/h =7. Determine f(0) and find the expression for f(x). (Hint: find f'(x)).

### 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?

### If the marginal profit function is positive at x=a, then it makes sense to increase the level of production.

Determine whether the statement is true or false. If it is true, explain why it is true. If false, give an example to show it is false. 1. If the marginal profit function is positive at x=a, then it makes sense to increase the level of production. 2. If f and g are differentiable and f(x)g(y) = 0, then dy/dx = - f

### 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.

### Implicit Differentiation Example Problems

Find dy/dx by implicit differentiation. 1. (2x+3y)^1/3 = x^2 2. x^2y^1/2 = x + 2y^3 3. The demand function for a certain make of ink jet cartridge is p= -0.01x^2 - 0.1x + 6 Where p is the unit price in dollars and x is the quantity demanded each week, measure in units of a thousand. Compute the elasticity of

### Partial derivatives word problem

The voltage V in volts across a fixed resistance r in series with a variable resistance R is V=(rE)/(R+r) where E is the source voltage. calculate the rate at which V changes with respect to time if when E=10 volts, R=12 ohms, and r=8 ohms, the source voltage is increasing at 2 volts per minute and the variable

### Directional derivative

Let f(x,y,z)=e^(x+y+3z) a) find the directional deriv at p(-2,2,1)in the direction of q(18,-2,4). b) find the direction of max increase at p. what is max rate of increase? c) find the direction of max decrease at p. what is max rate of decrease? d) in what direction from p will f remain constant? (multiple corr

### Calculus: Derivatives Example

Find the point(s) on the graph where the tangent line is horizontal... 50. Extend the product rule for differentiation to the following case involving the product of three differentiable functions... Find derivative... (Please see attached.)