1. Find dy/dx by implicit differentiation. Please show work in detail. 1/x^4 - 1

Question

1. Find dy/dx by implicit differentiation. Please show work in detail.

1/x^4 - 1/y^4 = 2

2. Find the first and second derivatives of the following: Please show work in detail.

f(x)=(x^2-6)^5

f(x)=2x(3x-6)^5

3. Consider the function f(x) = (3x-1)^6. There are brackets around whole problem raised to the 6th power. show work in detail

                                                  (5x+1)

     a. Find first derivative

     b. Find the equation of the tangent line at x=0

4. The altitude in feet of a vertically launched rocket t seconds after launch from the roof of a building is given by

     h(t)=1/12 t^4 - 2/3 t^3 + 4t^2+8t+5 where (0< or equal to t < or equal to 8). Find the acceleration, the velocity, and the altitude of the rocket 3 seconds after launch.

5. Determine the elasticity of demand function for a certain DVD, given the demand equation and the current rental price of a DVD where x is # of DVD's and p is the price in $. x=2/3sqrt(36-p^2). (0< or equal to p < or equal to 6). I need help finding the elasticity function.

6. Find the first derivative of 4xsqrt(2x^2+2x-1)

7. Determine the equation of the tangent line in slope 3 + 2x^2=1 at x=1.

8. Consider the function f(x)=2x/(x+2)^2:

     a. Showing your calculus and sign graph, find the intervals where it is increasing and decreasing.

     b. Find the intervals where it is concave up and concave down.

9. Find the relative maximam and relative minima, if any, of f(x) -1/4x^4 at 2x^2/ Show your calculus. Use either the first or second derivative test to justify your answer.

10. Suppose the second derivative of f(x)=4x^3-2x^2, and f(x) has critical numbers at x=-1 and x=1. Use the second derivative test to determine if any of the critical numbers gives a relative maximum.

11. Determine the intervals where the following functions ar18xe concave up and concave down. Also determine if there are any points of inflection. Show your calculus and sign graph.

a) g(x)=2x^3-6x^2-18x

b) f(x)=x/x+4

I know this is a lot but out of 100 questions these are the ones I cannot figure out. I need to be able to show my work in detail so I can understand for future reference.

Thank you! Any help is greatly appreciated.

Explanation / Answer

FOLLOW THIS

Welcome back same notation as before this time is an x on the end :P

Df=6x^2-12x-18
Even though I see the solution we'll use the quadratic formula (which I hope you know by now) to find the roots.
a=6, b=-12, c=-18 (if you don't know it send me a message or go to wolfram's for a guide as to how it works)

6*(x-3)*(x+1)=0 after all is said and done. Clearly we can remove the 6 without causing any problems
x=3 and -1 are the critical values of x.

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