[tex] \large \bold{SOLUTION:} [/tex]
[tex] \footnotesize \begin{array}{l} f(x) = x\sqrt{a^2 + x^2} + a^2 \arcsin\left(\dfrac{x}{a}\right) \\ \\ f'(x) = \dfrac{x}{\cancel{2}} (a^2 + x^2)^{-\frac{1}{2}} (\cancel{2}x) + \sqrt{a^2 + x^2} + a^2 \cdot \dfrac{1}{\sqrt{a^2 - x^2}} \\ \\ \boxed{f'(x) = \dfrac{x^2}{\sqrt{a^2 + x^2}} + \sqrt{a^2 + x^2} + \dfrac{a^2}{\sqrt{a^2 - x^2}}} \\ \\ \textsf{Note: } \\ \\ \quad d(uv) = u dv + v du \\ \\ \quad d\left[\arcsin \left(\dfrac{u}{a}\right)\right] = \dfrac{du}{\sqrt{a^2 - u^2}} \end{array} [/tex]
[tex] \blue{\mathfrak{\#CarryOnLearning}} [/tex]
