Practice Gateway Exam 5

Section 1.5 Practice Gateway Exam 5

The use of a calculator or any computer algebra system is not permitted on this test. No partial credit is available. A score of at least 7 correct is required to pass this gateway.

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Find the derivative of each of the following functions:

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Exercises Exercises

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1.

$y = 7 x^{6} + 2 x^{3.1} - 4 x + \frac{5}{x^{6}}$

Solution.

\begin{aligned} \frac{d y}{d x} & = \frac{d}{d x} [7 x^{6} + 2 x^{3.1} - 4 x + 5 x^{- 6}] \\ = 42 x^{5} + 6.2 x^{2.1} - 4 - 30 x^{- 7} \end{aligned}

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2.

$g (x) = x^{4} \ln (6 x^{5}) + 10 e^{3}$

Solution.

\begin{aligned} g^{'} (x) & = x^{4} \cdot \frac{d}{d x} [\ln (6 x^{5})] + \ln (6 x^{5}) \cdot \frac{d}{d x} [x^{4}] \\ = x^{4} \cdot \frac{1}{6 x^{5}} \cdot \frac{d}{d x} [6 x^{5}] + \ln (6 x^{5}) \cdot 4 x^{3} \\ = x^{4} \cdot \frac{1}{6 x^{5}} \cdot 30 x^{4} + \ln (6 x^{5}) \cdot 4 x^{3} \end{aligned}

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3.

$r (θ) = 8 (\sin (4 θ))^{- 10}$

Solution.

\begin{aligned} r^{'} (θ) & = - 80 (\sin (4 θ))^{- 11} \cdot \frac{d}{d θ} [\sin (4 θ)] \\ = - 80 (\sin (4 θ))^{- 11} \cdot \cos (4 θ) \cdot \frac{d}{d θ} [4 θ] \\ = - 80 (\sin (4 θ))^{- 11} \cdot \cos (4 θ) \cdot 4 \end{aligned}

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4.

$f (x) = \ln (6) 7^{100 x}$

Solution.

\begin{aligned} f^{'} (x) & = \ln (6) \cdot \ln (7) \cdot 7^{100 x} \cdot \frac{d}{d x} [100 x] \\ = \ln (6) \cdot \ln (7) \cdot 7^{100 x} \cdot 100 \end{aligned}

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5.

$f (w) = \frac{w^{6}}{\arcsin (w)}$

Solution.

\begin{aligned} f^{'} (w) & = \frac{\arcsin (w) \cdot \frac{d}{d w} [w^{6}] - w^{6} \cdot \frac{d}{d w} [\arcsin (w)]}{(\arcsin (w))^{2}} \\ = \frac{\arcsin (w) \cdot (6 w^{5}) - w^{6} \cdot \frac{1}{\sqrt{1 - w^{2}}}}{(\arcsin (w))^{2}} \end{aligned}

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6.

$g (t) = \arctan (7 t^{4})$

Solution.

\begin{aligned} g^{'} (t) & = \frac{1}{1 + (7 t^{4})^{2}} \cdot \frac{d}{d t} [7 t^{4}] \\ = \frac{1}{1 + (7 t^{4})^{2}} \cdot 28 t^{3} \end{aligned}

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7.

$h (z) = \cos (60 z^{2} + 100 z)$

Solution.

\begin{aligned} h^{'} (z) & = - \sin (60 z^{2} + 100 z) \cdot \frac{d}{d z} [60 z^{2} + 100 z] \\ = - \sin (60 z^{2} + 100 z) \cdot (120 z + 100) \end{aligned}

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8.

$h (x) = 6 e^{5 x} + \sqrt{x^{5} - 6 x}$

Solution.

\begin{aligned} h^{'} (x) & = \frac{d}{d x} [6 e^{5 x} + (x^{5} - 6 x)^{1 / 2}] \\ = 6 e^{5 x} \cdot \frac{d}{d x} [5 x] + \frac{1}{2} (x^{5} - 6 x)^{- 1 / 2} \cdot \frac{d}{d x} [x^{5} - 6 x] \\ = 6 e^{5 x} \cdot 5 + \frac{1}{2} (x^{5} - 6 x)^{- 1 / 2} \cdot (5 x^{4} - 6) \end{aligned}