derivative of θtanθ secθ

Just use the product rule

f(θ) = θtanθ secθ = u*v
f'(θ) = u'v + uv'
= 1tanθ secθ + x(tanθ secθ)'
= tanθ secθ + x[sec^2(θ)sec(θ) + tan(θ)sec(θ)tan(θ)]
= tan(θ)sec(θ) + x(sec^3(θ) + sec(θ)tan^2(θ)]
= tan(θ)sec(θ) + x(sec^3(θ) + sec(θ)(sec^2(θ)-1)]
= tan(θ)sec(θ) + x(2sec^3(θ) - sec(θ))]
= sec(θ)[tan(θ) + x(2sec^2(θ) - 1)]
= sec(θ)[tan(θ) + x(2tan^2(θ) + 1)]