urgent:trigonometric equation help and a few other questions

[lhl_anecar]

Can someone please kindly help me solve these questions

1. state the standard identity relating cos and sin arising from Pythagoras Theorem
    sec^2(x) + cosec^2(x) = sec^2(x) x cosec^2(x)

I have no idea what the question wants.

and this question

2. ln x + ln(1-x)-2ln(1+x)

It looks so simple but i just cant seem to get it right.. help
 
3. 5 x (1.5)^a = 100

how do i get a?

4. find the deriavative of the following

a) y=x^5e^x + 3xlnx

b. R=(e^t+1)(t-ln t)

y=(x^5)(e^x) + 3xlnx    how do i solve this??

[Freaked12]

what do you have to do
 simplify ?
 or equate ?

if we simplify
its coming
 ln[ (x-x*) / (1+x)* ]
 x* means x squared

and if we equate

 then you remove the ln
 by making the other side the power of e
 and equation it to the thing iside the ln bracket
 that will make a quadratic equation in x

[Freaked12]

just remember
if lnx - lnx
then its equal to ln [ x/x ]
and if lnx + lnx then it equals ln [x multiplied by x ]

[ashish]

Can someone please kindly help me solve these questions

1. state the standard identity relating cos and sin arising from Pythagoras Theorem
    sec^2(x) + cosec^2(x) = sec^2(x) x cosec^2(x)

I have no idea what the question wants.

the equation is
cosx² + sinx² = 1

for the second one i have to prove ?

secx is 1/cosx  and cosecx is 1/sinx

from left hand side

sec²x + cosec²x = 1/cos²x + 1/sin²x
         expressing as a single fraction
 
                                                      =(sin²x + cos²x)/ cos²x sin²x
                                                       = 1/sin²x   *   1/cos²x

                                                        = cosec²x  *  sec²x

[ashish]

Can someone please kindly help me solve these questions



It looks so simple but i just cant seem to get it right.. help
 
3. 5 x (1.5)^a = 100

how do i get a?

3.5 * (1.5)^a =100
          1.5^a= 100/3.5
         taking ln on both side
         ln1.5^a= ln(100/3.5)
                  power becomes multiplier (one property of log)
                  a=(ln(100/3.5))/ln1.5
                   a =8.268

[ashish]

Can someone please kindly help me solve these questions



4. find the deriavative of the following

a) y=x^5e^x + 3xlnx

b. R(e^t+1)(t-ln t)

4.a
let do it part by part

y=X<sup>5ex

dy</sup>/_dx = X^5ex-1 *5e^x


second part using quotient rule

U^dv/_dx + V^du/_dx
= 3x*(1/x) + 3lnx
=3+3lnx

add these two

5e^xX^5ex-1 + 3 + 3lnx

i just hope i didn't do any mistake with superscript 

[ashish]

for 4b i think you missed something 

[lhl_anecar]

thanks for the help!! appreciate it loads.. it's really kind of you guys to help me out.
i just have one last question that i am still unsure about and hope u guys can help me out
 which is

using the product rule to find the derivatives of the following..

a)  x^2[?x + (1/?x)]


thanks again!!

[lhl_anecar]

for 4b i think you missed something  

R=(e^t+1)(t-lnt)  

i missed out the = sign, sorry about that,i just don't seem to get how to differentiate ln. please help. thanks!!!

[ashish]

oops sorry hey i have to go to school for exams ,,   when i will do it when i will be back or maybe someone here will do it in the mean time 

[Deadly_king]

thanks for the help!! appreciate it loads.. it's really kind of you guys to help me out.
i just have one last question that i am still unsure about and hope u guys can help me out
 which is

using the product rule to find the derivatives of the following..

a)  x^2[?x + (1/?x)]


thanks again!!

First you need to expand the brackets and replace square roots by power 1/2.

Let y = x²(x^1/2 + 1/x^1/2)

Since they all have the same base(x), multiplication implies addition of powers while division implies subtraction of powers

y = x^(2+1/2) + x^(2-1/2) ----> y = x^5/2 + x^3/2

From here it's easy to differentiate : dy/dx = 5/2 x^3/2 + 3/2 x^1/2

The required solution can be simplifies as (5x^3/2 + 3x^1/2)/2

Hope it helps

[Deadly_king]

R=(e^t+1)(t-lnt)  

i missed out the = sign, sorry about that,i just don't seem to get how to differentiate ln. please help. thanks!!!

Is it R = (e^(t+1))(t - lnt) or R = (e^t + 1)(t - lnt)

[lhl_anecar]

Is it R = (e^(t+1))(t - lnt) or R = (e^t + 1)(t - lnt)

R = (e^t + 1)(t - lnt) it is this one..   sorry about the typing error. thanks!!!

[Freaked12]

well we can use the product rule

u = e^t + 1
u' = e^t

v = t - ln t
v' = 1 - 1/t

dR/dt = u'v + uv'
= e^t (t - ln t) + (e^t +1).(1-1/t)

Tell me if this is wrong.

[lhl_anecar]

y=(x^5)(e^x) + 3xlnx

how do i solve this?? i made a mistake typing the previous question... my bad, made the whole question more difficult..
someone..please help me out, again.. thanks!! thank you very very much......