#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)#
#"#
#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)#
#"#
#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)#
