1. f(x) = cos(x) - 4 tan(x)
Use the difference rule of differentiation:
Rightarrow f'(x) = frac(d)(dx)(cos(x)) - frac(d)(dx)(4 tan(x))
Rightarrow f'(x) = - sin(x) - 4 sec^(2)(x)
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2. f(x) = - 6 cos(x) + 2 tan(x)
Use the sum rule of differentiation:
Rightarrow f'(x) = frac(d)(dx)(- 6 cos(x)) + frac(d)(dx)(2 tan(x))
Rightarrow f'(x) = 6 sin(x) + 2 sec^(2)(x)
Substitute frac(3 pi)(4) in place of x:
Rightarrow f'(frac(3 pi)(4)) = 6 sin(frac(3 pi)(4)) + 2 sec^(2)(frac(3 pi)(4))
Rightarrow f'(frac(3 pi)(4)) = 6 cdot frac(sqrt(2))(2) + 2 cdot (- sqrt(2))
Rightarrow f'(frac(3 pi)(4)) = 3 sqrt(2) - 2 sqrt(2))
Rightarrow f'(frac(3 pi)(4)) = sqrt(2)
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3. f(x) = 7 sec(x)
Rightarrow f'(x) = 7 sec(x) tan(x)
Use the product rule:
Rightarrow f''(x) = 7 (sec(x) cdot frac(d)(dx)(tan(x)) + tan(x) cdot frac(d)(dx)(sec(x)))
Rightarrow f''(x) = 7 (sec(x) cdot sec^(2)(x) + tan(x) cdot sec(x) tan(x))
Rightarrow f''(x) = 7 (sec^(3)(x) + sec(x) tan^(2)(x))
Rightarrow f''(x) = 7 sec^(3)(x) + 7 sec(x) tan^(2)(x)
