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Whoops.  I forgot to put to the third power on (t^4-1).  The equation should be h(t)=(t^4-1)^3(t^3+1)^4.  Can you please do it again?  I am so sorry.

Find the derivative of the following via implicit differentiation:

d/dt(h(t)) = d/dt((1 + t^3)^4 (-1 + t^4))

The derivative of h(t) is h'(t):

h'(t) = d/dt((1 + t^3)^4 (-1 + t^4))

Use the product rule, d/dt(u v) = v ( du)/( dt) + u ( dv)/( dt), where u = (t^3 + 1)^4 and v = t^4 - 1:

h'(t) = (t^4 - 1) d/dt((1 + t^3)^4) + (t^3 + 1)^4 d/dt(-1 + t^4)

Using the chain rule, d/dt((t^3 + 1)^4) = ( du^4)/( du) ( du)/( dt), where u = t^3 + 1 and d/( du)(u^4) = 4 u^3:

h'(t) = (1 + t^3)^4 (d/dt(-1 + t^4)) + (-1 + t^4) 4 (t^3 + 1)^3 d/dt(1 + t^3)

Differentiate the sum term by term:

h'(t) = (1 + t^3)^4 (d/dt(-1 + t^4)) + d/dt(1) + d/dt(t^3) 4 (1 + t^3)^3 (-1 + t^4)

The derivative of 1 is zero:

h'(t) = (1 + t^3)^4 (d/dt(-1 + t^4)) + 4 (1 + t^3)^3 (-1 + t^4) (d/dt(t^3) + 0)

Simplify the expression:

h'(t) = 4 (1 + t^3)^3 (-1 + t^4) (d/dt(t^3)) + (1 + t^3)^4 (d/dt(-1 + t^4))

Use the power rule, d/dt(t^n) = n t^(n - 1), where n = 3.

d/dt(t^3) = 3 t^2:

h'(t) = (1 + t^3)^4 (d/dt(-1 + t^4)) + 3 t^2 4 (1 + t^3)^3 (-1 + t^4)

Simplify the expression:

h'(t) = 12 t^2 (1 + t^3)^3 (-1 + t^4) + (1 + t^3)^4 (d/dt(-1 + t^4))

Differentiate the sum term by term:

h'(t) = 12 t^2 (1 + t^3)^3 (-1 + t^4) + d/dt(-1) + d/dt(t^4) (1 + t^3)^4

The derivative of -1 is zero:

h'(t) = 12 t^2 (1 + t^3)^3 (-1 + t^4) + (1 + t^3)^4 (d/dt(t^4) + 0)

Simplify the expression:

h'(t) = 12 t^2 (1 + t^3)^3 (-1 + t^4) + (1 + t^3)^4 (d/dt(t^4))

Use the power rule, d/dt(t^n) = n t^(n - 1), where n = 4.

d/dt(t^4) = 4 t^3:

h'(t) = 12t^2 (1 + t^3)^3 (-1 + t^4) + 4t^3 (1 + t^3)^4 [Courtesy of Mathematica 11 Home Edition]

Thank You So Much For Your Help!!

Let the price of a movie = M

Let the price of a game = G

2M + 3G =24.30...............(1)

3M + G  =18.25...............(2)

Use substitution:

G = $5.20 - price of 1 game.

M = $4.35 - price of 1 movie.

Melody:

The smallest N is 6. The LCM of[12, 18, 24, 30, 60] = 360. Therefore:

N = 360k + 6, where k = 0, 1, 2, 3.........etc.

The whole second part of the second sentence wouldn't load onto the screen, until I replaced some of the variables with LATEX. sry

I think that a part of your question is missing.

Hi Kennedy,

I really do not understand your question.

I do not understand the estimation method that you have described and ..

Why would you use any estimation process for this?

Plus is easy and the numbers are not very big.

So I know I am not answering your question   Sorry 

A typical brick is a rectangular prism with dimensions 4 in. x 2 in. x 8 in.

A typical bathtub is a rectangular prism with dimensions 60 in. x 30 in. x 18 in.

60/4=15      

30/2=15

18/8=2.25

The base layer is 15*15=225 bricks

There are 2.25 layers so   225*2.25=506.25 = 507 bricks.

alternatively

4*2*8=64

60*30*18=32400

32400/64 = 506.25 = 507 bricks

Well ask your questions one at a time.

I have told you that before!

Learn from the answer, try the next one yourself and then ask the next one if you need to. 

Also when ever possible, which is just about always, type your question in. If there are diagrams then post them in as pictures.

I have told you that before too.

The smallest N is 6, the rest of my answer was incorrect. 

The figure ABCD has four sides, and its vertices lie on a circle. This situation has a specific name: cyclic quadrilateral. This situation also has a specific theorem associated to it: Opposite angles of a cyclic quadrilateral are supplementary. Let's use this theorem to our advantage:

Hey Guest!

\(\begin{align*} N &\equiv 6 \pmod{12}, \\ N &\equiv 6 \pmod{18}, \\ N &\equiv 6 \pmod{24}, \\ N &\equiv 6 \pmod{30}, \\ N &\equiv 6 \pmod{60}. \end{align*}\)

LCM [12, 18, 24, 30, 60] = 360

N = 360k + 6. 

6 is your answer.

Guest is right, my previous answer was wrong. 

I hope this helped,

gavin

Lol no ones going to give you all the answers to a test/quiz. At least try to do them yourself then ask for help on the ones you don't understand. 

The epicenter is (1,2)

Your Welcome!!!

To find  f( g(3) ) , first let's find  g(3) .

Since  g(3) = 1 ,  we can substitute  1  in for  g(3) .

f( g(3) )  =  f( 1 )

Now we just have to find  f(1) .

So....

f( g(3) )  =  f(1)  =  -1

f( g(3) )  =  -1

The area of the polygon is the number of little squares that fit into the polygon.

We can split the polygon up like this:

area of polygon  =  area of rectangle + area of triangle

area of rectangle  =  base * height

area of rectangle  =  9 * 2

area of rectangle  =  18        (You can count the squares to get the same answer.)

area of triangle  =  (1/2) * base * height

area of triangle  =  (1/2) * 9 * 6

area of triangle  =  27

area of polygon  =  area of rectangle + area of triangle

area of polygon  =  18 + 27

area of polygon  =  45

There are  45  square units in the polygon.

If you want ONLY distinct permutations, then you can have:

5! / 3! . 2! =10 distinct permutations as follows:

{2, 2, 2, 9, 9} | {2, 2, 9, 2, 9} | {2, 2, 9, 9, 2} | {2, 9, 2, 2, 9} | {2, 9, 2, 9, 2} | {2, 9, 9, 2, 2} | {9, 2, 2, 2, 9} | {9, 2, 2, 9, 2} | {9, 2, 9, 2, 2} | {9, 9, 2, 2, 2} (total: 10)

a) 7/10[d +5/6] = 7,                 multiply both sides by 10 / 7

[d + 5/6] = 10

[6d + 5] / 6 = 10                  cross multiply

6d + 5 = 60                         subtract 5 from both sides

6d =55                                divide both sides by 6

d = 55 / 6

b) 

5c/4 - 2c/3 =7/10              Put the LHS over the CD=12

[15c - 8c] /12 = 7/10

7c / 12 = 7/10                   cross multiply

7c = 84/10                        divide both sides by 7

c =84 / 10 x 1/7

c = 84 / 70                        divide both sides by 14

c = 6/5

S = F/ [1 - R], where S = sum of infinite series, F = First term, R = Common ratio

S = 4 / [1 - 6/7]

S = 4 / (1/7)

S = 4 x 7

S = 28 =7 + 6 + 5 + 4 + 3 + 2 + 1 + 0

Independant

P(A l B) = P(A)