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Section 3.3

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1. (a) f'(x) = 3x^2 + 6

(c) g'(t) = 3 - 12t

(e) f'(t) = 18t - 36

2. (a) f'(x) = 8x + 4

(c) \displaystyle{g'(x) = \frac{11}{(2x + 5)^2}}

(e) \displaystyle{f'(t) = \frac{6t^6 + 104t^3 - 6t^2 - 48}{(2t^3 + 6)^2}}

(g) h'(t) = -\dfrac{3}{t^2}

(i) \displaystyle{h'(z) = 24z^2 + \frac{1}{2z^2}}

3. (a) \dfrac{ds}{dt} = (t^2 - 6t +3)(32t^3 + 12t) + (8t^4 + 6t^2 - 7)(2t - 6)

(c) \dfrac{dy}{dx} = (x^2 - 2x + 3)(2x^2 + 13x - 6)(6x - 4)
+ (x^2 - 2x + 3)(4x + 13)(3x^2 - 4x + 1)
+ (2x - 2)(2x^2 + 13x - 6)(3x^2 - 4x + 1)

4. (a) k'(2) = 26

(c) k'(2) = 102

5. (a) \displaystyle{v(t) = -1 + \frac{1}{2}t^2}

(c) The object is at the origin when t = -\sqrt{6}, 0, \sqrt{6} . The velocities are: v(-\sqrt{6}) = 2 , v(0) = -1 , v(\sqrt{6}) = 2 .

(e) -\sqrt{2} < t < \sqrt{2}

(g) a(t) = t

(i) The object is moving toward the left and slowing down at time t = 1.

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