### Derivatives

Please show how to solve to the answer of 1/2 If f(x) = x-1/x +1 for all x not equal to -1, then f'(1) =

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Please show how to solve to the answer of 1/2 If f(x) = x-1/x +1 for all x not equal to -1, then f'(1) =

If y = 4x + 3 is tangent to the curve y = x^2 + k, then k is: Please show how to solve to the answer of 7

Water is running out of a conical funnel at the rate of 1 cubic inch per second. If the radius of the top of the funnel is 4 inches and the height is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top.

The curve y = x^3 + x^2 - x has two horizontal tangents. Find the distance between these two tangents and draw a picture. Please show how to solve the following problem to the answer of 32/27.

If u = x2 +y2 and v = 2xy, transform F(u,v) into G(x,y). Find δG/δx and δG/δx in terms of δF/δu and δG/δy. Please see the attached file for the fully formatted problem.

Y=(x+1)/(x^2-4) 1) Find dy/dx and second derivative 2) What are the asymptotes? 3) Intervals of increase? intervals of upward concavity? Points where the horizontal line is tangent?

When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the equation PV^1.3 = k where k is a constant. Find the relationship between the related rates dp/dt and dV/dt Answer: V^0.3 (1.3p dV/dt) + V dp/dt = 0

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately 3x the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?

Please verify each differentiation formula and explain how. (a) d/dx[arctan u] = u'/1 + u^2 (b) d/dx[arcsec u] = u'/lul (square root of u^2 -1) (c) d/dx[arccos u] = -u'/square root of 1 - u^2 (d) d/dx[arccot u] = -u'/1- u^2 (e) d/dx[arccsc u] = -u'/lul (square root of u^2 - 1

Please explain how to find dy/dx using logarithmic differentiation. Please solve to the designated answer. y = x to the 2/x power Answer: 2(1 - ln x)x^(2/x - 2) power

Please explain how to find d^2y/dx^2 in terms of x and y. Please solve to the designated answer. y^2 = x^3 Answer: 3x/4y

Please explain how to find the equations of the tangent line and the normal line to the graph of the equation at the indicated point and achieve the specified answer. y ln x + y^2 = 0 (e, -1)

Find dy/dx using logarithmic differentiation (a) y = x √(x^2 - 1) Answer: 2x^2 - 1/√(x^2 - 1) (b) y = x^2(√(3x - 2)/(x - 1)^2 Answer 3x^3 - 15x^2 + 8x/2(x - 1)^3 √(3x - 2) (c) y = x^2/x Answer : 2(1-ln x)x^2/x-2

Find the slope of the tangent line at the indicated point. (a) Witch of Agnessi (x^2 + 4)y = 8 Point: (2, 1) (b) Bifolium (x^2 + y^2)^2 = 4(x^2)y Point: (1,1)

Let u be a differentiable function of x. Use the fact that lul = the square root of u^2 to prove that d/dx[lul] = u'(u/lul). u is not equal to zero.

Air is being pumped into a spherical balloon so that the radius is increasing at the rate of dr/dt = 3 inches per second. What is the rate of change of the volume of the balloon in cubic inches per second, when r = 8 inches? Hint: V = 4/3(pi)r^3

Please explain how to solve to the following problem: A buoy oscillates in simple harmonic motion y = A cos omega(t) The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its

How do I find the anti derivative of the attached problem?

2 (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s how fast is the area of the spill increasing when the radius

(See attached file for full problem description with equations) --- real numbers complex numbers Suppose , are continuous functions with nonvanishing first partial derivatives. Let , Show that . ---

Differentiate: 1) y=(cos(x)^cot^2(x) 2) find the second derivative of sin(x^2*y)=-cos(x^2*y) note: *=multiply

The curve y = ax2(ax sqaured) + bx passes through the point (2, 4) with gradient 8. Find a and b I have the answer but do not understand how to get to it. Therefore, could you please show full workings. Many thanks.

The function has a derivative everywhere and has just one critical point, . In parts (a)-(d), you are given additional conditions. In each case decide whether is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases. a) b) c) d) Please see th

Assume that you collect P dollars from a transaction and being a mathematics wiz, you have developed formula to calculate the future value of your investment: where, r is the rate of interest and t is the time horizon. Suppose you invest your profit, P dollars, from above transaction, and invest it in a bank at 5% rate of

Let be a measurable space and let be two -finite measures defined on . Suppose and is the Radon-Nikodym derivative of with respect to . Define by Show that is a well-defined linear isometry and is an isomorphism if and only if (i.e are mutually absolutely continuous). ---

I am not sure how to set up this problem, I think that I have to use the exponential rule, but after that I am lost. (See attached file for full problem description with complete equations) --- The quantity q, of a certain skateboard sold depends on the selling price, p, in dollars, so, we write q = f(p). You are give

1).Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 AM. Assume that the pollutant disperses horizontally, form

Find the derivative of a) y=cuberoot(4x^2-sqrt(cotx)) this is referring to cuberoot of entire function b)ln(x^2)+(ln x)^2-ln(e^x^2)+e^x^2

Find the derivative of the following functions g(x)= ln(x-2/e^x-2) h(x)=tan^3(2x)/sin^3(2x)

A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints). Let p(x,y) = max_[0,1]|x'(t) - y'(t)|. 1).Show that ( M,p) fails to be a metric space. 2). Let p(x,y) = |x(0) - y(0)| + max_[0,1]|x'(t) - y'(t)|. Is (M,p) now a metric space? Please