Find the derivative of
f(x) = sin^2 (e^sin^2 x)
I wonder if I'm right
F'(x) = 2cos(e^sin^2 x)
(sin^2 x e^sin^2 x -1)
not quite
if f = sin^2(u)
f' = 2sin(u) cos(u) u'
since u = e^v
u' = e^v v'
f = sin^2(e^sin^2 x)
f' = 2 sin(e^sin^2 x)cos(e^sin^2 x) * e^(sin^2 x) * 2 sinx cos x
= sin2x sin(2e^sin^2 x) e^(sin^2 x)
thank you very much
To find the derivative of f(x) = sin^2(e^sin^2(x)), you can apply the chain rule. The chain rule states that if you have a composition of functions, such as f(g(x)), the derivative can be found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's break it down step by step:
Step 1: Identify the outer and inner functions.
In this case, the outer function is sin^2(u), where u = e^sin^2(x), and the inner function is u = e^sin^2(x).
Step 2: Find the derivative of the inner function.
The derivative of e^x is simply e^x. Since x = sin^2(x) for the inner function, we need to apply the chain rule again. The derivative of sin^2(x) is 2sin(x)cos(x). So, the derivative of the inner function is e^sin^2(x) * 2sin(x)cos(x).
Step 3: Find the derivative of the outer function.
The derivative of sin^2(u) can be found by using the chain rule. The derivative of sin^2(u) is 2sin(u) * cos(u).
Step 4: Multiply the derivatives from steps 2 and 3.
Multiply the derivative of the outer function by the derivative of the inner function:
2sin(u) * cos(u) * e^sin^2(x) * 2sin(x)cos(x).
Putting all these steps together, the derivative of f(x) = sin^2(e^sin^2(x)) is:
2sin(e^sin^2(x)) * cos(e^sin^2(x)) * e^sin^2(x) * 2sin(x)cos(x).
So your answer should be:
F'(x) = 4sin(x)cos(x) * sin(e^sin^2(x)) * cos(e^sin^2(x)) * e^sin^2(x)