\(\displaystyle Given: \ xtan(\theta) \ = \ 100, \ \frac{dx}{dt} \ = \ 8\)
\(\displaystyle Find \ \frac{d \theta}{dt} \ when \ x \ = \ 2 \ and \ \theta \ = \ \frac{\pi}{3}.\)
\(\displaystyle \frac{dx}{dt}tan(\theta)+xsec^{2}(\theta)\frac{d \theta}{dt} \ = \ 0\)
\(\displaystyle 8tan(\frac{\pi}{3})+2sec^{2}(\frac{\pi}{3}) \frac{d \theta}{dt} \ = \ 0\)
\(\displaystyle 8\sqrt3+8\frac{d \theta}{dt} \ = \ 0\)
\(\displaystyle Hence, \ \frac{d \theta}{dt} \ = \ \frac{-8\sqrt3}{8} \ = \ -\sqrt3\)
