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For questions 1 - 3, take the derivative.

1.  \(f(x) = \sin(x^2 + 2^x)\)

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

2.  \(f(x) = 2^{2x}\)

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.  \(f'(x) = (2 \ln 2) 2^{2x}\)

3.  \(h(x) = \frac{e^{(\sin x)}}{x}\)

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

4.  If \(\sin(xy) = y\), find \(\frac{dy}{dx}\)

a.  \(\frac{dy}{dx} = \frac{y\cos(xy)}{1-x\cos(xy)}\)

b.  \(\frac{dy}{dx} = \frac{y\cos(xy)}{1+x\cos(xy)}\)

c.  \(\frac{dy}{dx} = \frac{-y\cos(xy)}{1+x\cos(xy)}\)

d.  \(\frac{dy}{dx} = \cos(xy)\)

e.  \(\frac{dy}{dx} = y\cos(xy)\)

5.  Find \(\frac{dy}{dx}\) if \(x^3y + x + y^2 = 11\).

a.  \(\frac{dy}{dx} = \frac{-3x^2y-1}{x^3+2y}\)

b.  \(\frac{dy}{dx} = \frac{x^3+2y}{-3x^2y-1}\)

c.  \(\frac{dy}{dx} = 3x^2 + 1 + 2y\)

d.  \(\frac{dy}{dx} = 3x^2y + x^3 + 1 + 2y\)

e.  \(\frac{dy}{dx} = 3x^2y + 1 + 2y\)

6.  Find the equation of the tangent line to the curve \(x\sin y = 2 y\) at the point \( (\pi, \frac{\pi}{2}\)

a.  \( y = 2x - \frac{\pi}{2}\)

b.  \( y = 2x - \frac{3\pi}{2}\)

c.  \( y = \frac{1}{2} x - \frac{\pi}{2}\)

d.  \( y = \frac{1}{2} x \)

e.  \( y = x - \frac{\pi}{2}\)

7.  A spherical balloon is inflated with gas at the rate of 400 cubic centimeters per second (that is, the volume of the sphere is increasing at a rate of 400 cubic centimeters per second). How fast is the radius of the balloon increasing when the radius is 50 cm?
The volume of a sphere is \(\frac{4}{3}\pi\) times the radius cubed.

a.  \(\frac{0.02}{\pi}\) cm/s

b.  \(\frac{0.04}{\pi}\)cm/s

c.  \(\frac{0.2}{\pi}\) cm/s

d.  \(\frac{0.4}{\pi}\) cm/s

e.  2 \(\pi\) cm/s