derivatives:
2. cos teta/ 1+sin^2
To find the derivative of the given function, we will use the quotient rule.
The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then its derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Now let's apply this rule to find the derivative of the given function f(t) = cos(t) / (1 + sin^2(t)).
We have:
g(t) = cos(t)
h(t) = 1 + sin^2(t)
Now let's find the derivatives of g(t) and h(t):
g'(t) = -sin(t)
h'(t) = 2sin(t) * cos(t)
Plugging these values into the quotient rule, we get:
f'(t) = (g'(t) * h(t) - g(t) * h'(t)) / (h(t))^2
= (-sin(t) * (1 + sin^2(t)) - cos(t) * (2sin(t) * cos(t))) / (1 + sin^2(t))^2
To simplify this expression further, we can expand it:
f'(t) = (-sin(t) - sin^3(t) - 2sin(t) * cos^2(t)) / (1 + 2sin^2(t) + sin^4(t))
And that is the derivative of the given function f(t) = cos(t) / (1 + sin^2(t)).