How do you differentiate sin^2(cos t)
My work:
=2sin(cos)(cost)(-sint)
u= sin^2(z)
u'=2sin(z)dz/dt=2sin(cost)*(-sint)
and you are correct.
To differentiate sin^2(cos t), you can use the chain rule. The chain rule states that if you have a composite function, f(g(t)), the derivative with respect to t is given by the derivative of the outer function, f'(g(t)), multiplied by the derivative of the inner function, g'(t).
In this case, the outer function is f(u) = u^2, where u = sin(cos t), and the inner function is g(t) = cos t.
Let's find the derivative step by step:
1. Take the derivative of the outer function: f'(u) = 2u.
2. Take the derivative of the inner function: g'(t) = -sin t.
3. Apply the chain rule: (f(g(t)))' = f'(g(t)) * g'(t).
= 2u * g'(t)
= 2sin(cos t) * (-sin t)
Therefore, the derivative of sin^2(cos t) with respect to t is -2sin(cos t) * sin t.