# Question 5c963

Jul 16, 2017

See explanation below.

#### Explanation:

$1. f \left(x\right) = \cos \left(x\right) - 4 \tan \left(x\right)$

Use the difference rule of differentiation:

$R i g h t a r r o w f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\cos \left(x\right)\right) - \frac{d}{\mathrm{dx}} \left(4 \tan \left(x\right)\right)$

$R i g h t a r r o w f ' \left(x\right) = - \sin \left(x\right) - 4 {\sec}^{2} \left(x\right)$

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$2. f \left(x\right) = - 6 \cos \left(x\right) + 2 \tan \left(x\right)$

Use the sum rule of differentiation:

$R i g h t a r r o w f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(- 6 \cos \left(x\right)\right) + \frac{d}{\mathrm{dx}} \left(2 \tan \left(x\right)\right)$

$R i g h t a r r o w f ' \left(x\right) = 6 \sin \left(x\right) + 2 {\sec}^{2} \left(x\right)$

Substitute $\frac{3 \pi}{4}$ in place of $x$:

$R i g h t a r r o w f ' \left(\frac{3 \pi}{4}\right) = 6 \sin \left(\frac{3 \pi}{4}\right) + 2 {\sec}^{2} \left(\frac{3 \pi}{4}\right)$

$R i g h t a r r o w f ' \left(\frac{3 \pi}{4}\right) = 6 \cdot \frac{\sqrt{2}}{2} + 2 \cdot \left(- \sqrt{2}\right)$

Rightarrow f'(frac(3 pi)(4)) = 3 sqrt(2) - 2 sqrt(2))

$R i g h t a r r o w f ' \left(\frac{3 \pi}{4}\right) = \sqrt{2}$

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$3. f \left(x\right) = 7 \sec \left(x\right)$

$R i g h t a r r o w f ' \left(x\right) = 7 \sec \left(x\right) \tan \left(x\right)$

Use the product rule:

$R i g h t a r r o w f ' ' \left(x\right) = 7 \left(\sec \left(x\right) \cdot \frac{d}{\mathrm{dx}} \left(\tan \left(x\right)\right) + \tan \left(x\right) \cdot \frac{d}{\mathrm{dx}} \left(\sec \left(x\right)\right)\right)$

$R i g h t a r r o w f ' ' \left(x\right) = 7 \left(\sec \left(x\right) \cdot {\sec}^{2} \left(x\right) + \tan \left(x\right) \cdot \sec \left(x\right) \tan \left(x\right)\right)$

$R i g h t a r r o w f ' ' \left(x\right) = 7 \left({\sec}^{3} \left(x\right) + \sec \left(x\right) {\tan}^{2} \left(x\right)\right)$

$R i g h t a r r o w f ' ' \left(x\right) = 7 {\sec}^{3} \left(x\right) + 7 \sec \left(x\right) {\tan}^{2} \left(x\right)$