college level calculus

Question

pls follow the Chain Rule

d/dθ (5−12 ln | cos 3θ | )

Answer 1

Hi the problem, d/dθ (5−12 ln | cos 3θ | ) is same as d/dθ (12 ln | cos 3θ | ) same as -12*d/dθ ( ln | cos 3θ | )

cos 3θ> 0, derivative is -12*(-sin(3θ)*3/cos 3θ)=+36tan 3θ

cos 3θ< 0, derivative is -12*(sin(3θ)*3/-cos 3θ)=+36tan 3θ

cos3θ=0 means 3θ=+/-π2 + 2πk, k=0.1.-1.2,-2,...

θ=+/-π6 + 2πk/3, k=0.1.-1.2,-2,...

at θ=+π6 in the derivative approaches +infinity matters when θ approaches π6 from left (cos is less when theta is more) and -inifiity when θ approaches π6 from right. Same is true of all cos3θ=0 means 3θ=+/-π2 + 2πk, k=0.1.-1.2,-2,..

Answer 2

f(θ) = 5-12ln|cos(3θ)|

f'(θ) = -12(1/cos(3θ))(-sin(3θ))(3)

= 36sin(3θ)/cos(3θ)

= 36tan(3θ)