Respuesta :
Given the function f(x) defined as:
[tex]f(x)=3\tan(2x)[/tex]We need to find the derivative first. Using the chain rule, we know that:
[tex](\tan u)^{\prime}=u^{\prime}\cdot\sec²u[/tex]Then, taking the derivative if u = 2x:
[tex]\begin{gathered} f^{\prime}(x)=3(2)\sec²(2x) \\ \\ \Rightarrow f^{\prime}(x)=6\sec²(2x) \end{gathered}[/tex]Using this result, we can evaluate the derivative at x = π/2:
[tex]\begin{gathered} f^{\prime}(\frac{\pi}{2})=6\sec²(2\cdot\frac{\pi}{2})=6\sec²(\pi)=6\cdot(-1)² \\ \\ \therefore f^{\prime}(\frac{\pi}{2})=6 \end{gathered}[/tex]