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For questions 1 - 5, take the derivative.

1.  \(f(x) = \ln(\ln(\ln x ))\)

a.  \( f'(x) = \frac{1}{x^3}\)

b.  \( f'(x) = \frac{1}{(\ln x)^3}\)

c.  \( f'(x) = \frac{x}{(\ln x)^3}\)

d.  \( f'(x) = \frac{1}{x (\ln x)(\ln(\ln x))}\)

e.  \( f'(x) = \frac{\ln x}{x^3}\)

2.  \(g(x) = \sin(\pi x)\)

a.  \(g'(x) = \cos(\pi x)\)

b.  \(g'(x) = \cos(\pi)\)

c.  \(g'(x) = x\cos(\pi x)\)

d.  \(g'(x) = \pi \cos(\pi x)\)

e.  \(g'(x) = 0\)

3.  \(f(x) = \sin(x^2 + 2^x)\)

a.  \( f'(x) = \cos(x^2+2^x)\)

b.  \( f'(x) = \cos(x^2+2^x)(2x + (\ln 2)2^x)\)

c.  \( f'(x) = \cos(2x + x2^{(x-1)})\)

d.  \( f'(x) = \cos(2x + (\ln 2)2^x)\)

e.  \( f'(x) = \sin(x^2 + 2^x)(2x + x 2^{(x-1)})\)

4.  \(f(x) = 2^{2x}\)

a.  \(f'(x) = (2x) 2^{(2x - 1)}\)

b.  \(f'(x) = (2x) 2^{2x}\)

c.  \(f'(x) = x 2^{2x}\)

d.  \(f'(x) = (\ln 2) 2^{2x}\)

e.  \(f'(x) = (2 \ln 2) 2^{2x}\)

5.  \(h(x) = \frac{e^{(\sin x)}}{x}\)

a.  \(h'(x) = \frac{e^{(\sin x)}(\cos x) x - e^{(\sin x)}}{x^2}\)

b.  \(h'(x) = e^{(\sin x)}\)

c.  \(h'(x) = \frac{e^{(\sin x)} - e^{(\sin x)}(\cos x)x}{x^2}\)

d.  \(h'(x) = e^{(\sin x)} - e^{(\sin x)}(\cos x) x\)

e.  \(h'(x) = \frac{e^{(\sin x)} - e^{(\cos x)}x}{x^2}\)

In Problems 6 - 9 assume that the height, \(h\), of a hot-air balloon in meters, \(t\) minutes after it is launched is given by the function \( h(t) = t^3 - 8t^2 + 16 t\)

6.  What is the height of the balloon after 2 minutes?

a.  2 m

b.  4 m

c.  8 m

d.  16 m

e.  32 m