help to solve this complicated math question?

Question

find dy/dx if y=x-(1/2)(x^2+4) tan^-1 (x/2)
omg, this is so hard. this 1 question worth 50 points of the homework. help!!!

Answer 1

y = f(x) g(x) h(x)
product rule
dy/dx = f'gh + fg'h + fgh'
f'= -1/2x^(-3/2)
g'=2x
h'= 1/(1-(x^2)/4)^(1/2)

Answer 2

I don't know if you know the formula for the derivative of the arctan function, so let's derive that first.

Let y = arctan x, so tan y = x. Implicit differentiation gives us
sec^2 y dy/dx = 1
so dy/dx = 1/sec^2 y = 1 / (tan^2 y + 1) = 1 / (x^2 + 1).

So for y = x - (1/2) (x^2 + 4) arctan (x/2) we get
dy/dx = 1 - (1/2) (x^2 + 4) d/dx (arctan (x/2)) - (1/2) (2x) arctan (x/2)
= 1 - (1/2) (x^2 + 4) (1 / ((x/2)^2 + 1)) (1/2) - x arctan (x/2)
= 1 - (x^2 + 4) (1 / (4(x^2/4 + 1))) - x arctan (x/2)
= 1 - (x^2 + 4) (1 / (x^2 + 4)) - x arctan (x/2)
= 1 - 1 - x arctan (x/2)
= - arctan (x/2).

Answer 3

well, not very hard. but requires a lot of works.

dy/dx=1-(1/2)(x^2+4) d/dx [tan^-1 (x/2)] -x tan^-1 (x/2)

now, lets look at "d/dx [tan^-1 (x/2)]" from the equation above. If you know the Chain Rule, you use that to composite function tan^-1 (x/2). you have

d/dx [tan^-1 (x/2)]= {1 / [1+(x/2)^2]} * (1/2), ok, lets make it easier to see. lets write x/2 = 0.5x

d/dx [tan^-1 (x/2)]= {1 / [1+(0.5x)^2]} * (1/2) = 1 / (2+0.5x^2)

now, substitute. you have
dy/dx=1-(1/2)(x^2+4) [1 / (2+0.5x^2)] -x tan^-1 (x/2)
=1- [(x^2+4)/(x^2+4)] -x tan^-1 (x/2)
=1-1-x tan^-1 (x/2)
= -x tan^-1 (x/2)

i cant believe i actually did the whole thing!! i should have more than 10 points.

Answer 4

properly, it takes her 30 min, because of the fact the arms have been at the same time then precisely opposite, with a view to try this, one million/2 an hour could have previous. For the different section, in case you think of with reference to the tic marks on a a clock, there are 5 marks between hours (after the 4 to the 5.) So, each and each mark represents a 12 minute increment. Now, with a view to for the arms to begin on staggering of eachother, she would have mandatory to start a minimum of around 4: 20 (somewhat after fairly). it can't be on the 36 minute tick mark, b/c then this is too previous through end the only million/2 hour, so the initiating is around 4:24 and the top 4:fifty 4 yet you will could do the finer tuning your self ^^

Answer 5

my guess is 1

Answer 6

do the parentheses first then your on your own cause i dont know the rest.