Given the following functions:
[tex]\begin{gathered} f(u)=\cot(\frac{\pi u}{20}) \\ \\ u=g(x)=5\sqrt{x} \end{gathered}[/tex]
We find the composite function (f o g)(x):
[tex](f\circ g)(x)=f(g(x))=\cot(\frac{\pi g(x)}{20})=\cot(\frac{\pi\cdot5\sqrt{x}}{20})=\cot(\frac{\pi\sqrt{x}}{4})[/tex]
Now, we take the derivative of the composite function (using the chain rule):
[tex]\begin{gathered} (f\circ g)^{\prime}(x)=(\frac{\pi\sqrt{x}}{4})^{\prime}\cdot\cot^{\prime}(\frac{\pi\sqrt{x}}{4})=\frac{1}{2}\cdot\frac{\pi}{4\sqrt{x}}(-\csc^2(\frac{\pi\sqrt{x}}{4})) \\ \\ \\ (f\circ g)^{\prime}(x)=-\frac{\pi\csc^2(\frac{\pi\sqrt{x}}{4})}{8\sqrt{x}} \end{gathered}[/tex]
Finally, evaluating for x = 4:
[tex]\begin{gathered} (f\circ g)^{\prime}(4)=-\frac{\pi\csc^2(\frac{\pi\sqrt{4}}{4})}{8\sqrt{4}}=-\frac{\pi\csc^2(\frac{\pi}{2})}{16} \\ \\ \therefore(f\circ g)^{\prime}(4)=-\frac{\pi}{16} \end{gathered}[/tex]
