Need help with derivatives

[John45]

1) Find the derivative

y=(xsinx)/1+cosx

2) Find the second derivative

y=cosxsinx

3)

g(x)= x-sinx

a. Find all horizontal tangents of g(x)
b.Find where the slope of g(x) = 1

_________________________________________________
These are the answers that I got i feel that #1 is wrong

1.
(xcos(x)+cos^2(x)+(xsin^2(x)) / (1+cos(x))^2

2.
-sin^2(x) - cos^2(x)

3.
a) n(Pi/2), where n is any real odd integer
b) nPi, where n is any real odd integer

 

[Deleted member 4993]

1) Find the derivative

y=(xsinx)/1+cosx

2) Find the second derivative

y=cosxsinx

3)

g(x)= x-sinx

a. Find all horizontal tangents of g(x)
b.Find where the slope of g(x) = 1

_________________________________________________
These are the answers that I got i feel that #1 is wrong

1.
(xcos(x)+cos^2(x)+(xsin^2(x)) / (1+cos(x))^2

2.
-sin^2(x) - cos^2(x)

3.
a) n(Pi/2), where n is any real odd integer
b) nPi, where n is any real odd integer

(1)

and (2) are wrong

(3) I have not tried yet.

If you want us find the mistake - please post your work

 

[John45]

this is my work

1) (x sinx)/(1+cosx )

f(x) = x sin x
f^1(x) = x cos x
g(x) = 1 + cos x
g^1(x) = -sin x
(1 + cos x)(x cos x) – (x sin x)(-sin x)
x cos x + x ?cos?^2x – (-x?sin?^2x)
?(1+cos?x)??^2

x cos x + x ?cos?^2x + x?sin?^2x
?(1+cos?x)??^2




2) y = cos ? sin ?
f(x) = cos ?
f^1 = - sin ?
g(x) = sin ?
g^1(x) = cos ?

y^1 = cos ? (cos ?) + (-sin ?) (sin ?)

y^11 = - ?sin?^2 ? - ?cos?^2 ?

 

[Deleted member 4993]

this is my work

1) (x sinx)/(1+cosx )

f(x) = x sin x
f^1(x) = x cos x
g(x) = 1 + cos x
g^1(x) = -sin x
(1 + cos x)(x cos x) – (x sin x)(-sin x)
x cos x + x ?cos?^2x – (-x?sin?^2x)
?(1+cos?x)??^2

x cos x + x ?cos?^2x + x?sin?^2x
?(1+cos?x)??^2

I'm not exactly sure what you are doing - but one of the correct way to do this is as follows:

\(\displaystyle \frac{d}{dx}\left [\frac{x\ sin(x)}{1+cos(x)} \right ]\)

\(\displaystyle = \ \frac{[sin(x)+xcos(x)][1+cos(x)] - [-sin(x)][x\ sin(x)]}{[1+cos(x)]^2}\)

\(\displaystyle = \ \frac{[sin(x)+x \ cos(x) + sin(x) \ *\ cos(x)+x\ cos^2(x)] +[x\ sin^2(x)]}{[1+cos(x)]^2}\)

\(\displaystyle = \ \frac{sin(x)+x \ cos(x) + sin(x) \ *\ cos(x) \ + \ x}{[1+cos(x)]^2}\)

\(\displaystyle = \ \frac{[sin(x)\ +\ x] \ * \ [cos(x)\ +\ 1]}{[1+cos(x)]^2}\)

\(\displaystyle = \ \frac{sin(x)\ +\ x}{1+cos(x)}\)

done.....
2) y = cos ? sin ?

y = 0.5 * sin(2?)

y' = cos(2?) ... done

f(x) = cos ?
f^1 = - sin ?
g(x) = sin ?
g^1(x) = cos ?
y^1 = cos ? (cos ?) + (-sin ?) (sin ?) .................. Correct upto here
y^11 = - ?sin?^2 ? - ?cos?^2 ? ...... How did you get a negative sign for cos[sup:arxtn9cm]2[/sup:arxtn9cm]? here ???