Expert: Sherman D. - 10/22/2005

Question
find the derivatives of the functions.

1) h(x)=xtan(2sqrt(x))+7
2) f(x)=((sin x)/(1+cos x))^2

Answer
1.)
h(x) = xtan(2sqrt(x)) + 7
h(x) = xtan(2sqrt(x))
h(x) = xtan(2sqrt(x))
h(x) = x * tan(2sqrt(x))

Using the Product Rule

f(x)'g(x) + f(x)g(x)'
wheras
f(x) = x
g(x) = tan(2sqrt(x))

f(x)'g(x) = x^(1 - 1) * tan(2sqrt(x))
f(x)'g(x) = 1 * tan(2sqrt(x))
f(x)'g(x) = tan(2sqrt(x))

f(x)g(x)' = x * (sec(2sqrt(x)^2) * (2sqrt(x))')
f(x)g(x)' = x * (sec(2sqrt(x)^2) * (2(x)^(1/2)))
f(x)g(x)' = x * (sec(2sqrt(x)^2) * ((x)^((1/2) - 1)')
f(x)g(x)' = x * (sec(2sqrt(x)^2) * (1/(sqrt(x))))

h(x) = tan(2sqrt(x)) + ((xsec(2sqrt(x))^2)/(sqrt(x)))
h(x) = tan(2sqrt(x)) + (x/(sqrt(x)) * sec(2sqrt(x))^2
h(x) = tan(2sqrt(x)) + ((x^1)/(x^(1/2))) * sec(2sqrt(x))^2)
h(x) = tan(2sqrt(x)) + (x^(1 - (1/2)) * sec(2sqrt(x))^2)
h(x)' = tan(2sqrt(x)) + ((sqrt(x))sec(2sqrt(x))^2)

Just to let you know, tan(2sqrt(x)) isn't with ((sqrt(x))sec(2sqrt(x))^2)

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2.)
f(x) = (sin(x) / (1 + cos(x))^2
f(x) = (sin(x) / (1 + cos(x))^(2 - 1)
f(x) = sin(x)/(1 + cos(x))

derivative of sin(x) is cos(x)

derivative of 1 + cos(x) is -sin(x)

f(x)' = (-cos(x))/(sin(x))

Info found at http://people.hofstra.edu/faculty/Stefan_Waner/trig/trig3.html

If any of these are wrong, i shall report back to you, because i have asked someone else to see if they say the same thing or not.