Desperately need help with this derivative problem using chain rule?

Question

I've been doing my homework but am stuck on this chain rule problem. Any help would be stellar. Thanks, I love you all.

Let f(x) = e^x and g(x) = sin(x)
Find the derivatives of the following:

1) f(f(x))

2) g(g(x))

Answer 1

1)

Note that f(f(x)) = f(e^x) = e^(e^x).

d/dx e^(e^x) = e^(e^x) (e^x), which can be simplified to
d/dx e^(e^x) = e^(e^x + x)

2) g(g(x)) = g(sin(x)) = sin(sin(x))

d/dx sin(sin(x)) = cos(sin(x)) cos(x)

Answer 2

Question 1
f (x) = e^x and g(x) = sin x

Let h = f (f (x) = f (e^x) = e^(e^x)

h `(x) = e^(x). e^(e^(x))

Question 2
g(g(x)) = g(sin x) = sin ( sin x )

let h(x) = sin (sin x)

h `(x) = cos (sin x) . cos x

Answer 3

1. f' (f(x)) * f'(x) = (e^(e^x)) * (e^x)

2. cos (sin(x)) * cos(x), similarly

Answer 4

1)e^x.(e^e^x)
2)cosx.cos(sinx)




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10 pts. plz