This is should be a straightforward question for me but I'm blacking out right now.
Let $g$ be a differentiable function that satisfies $g(x) + x^3 \sin(g(x)) = x^4 + 4x$ around $x=1$.
If $g(1) = \frac{\pi}{2}$ , find the value of $g'(1)$.
Now, am I plugging in $\frac{\pi}{2}$ in all the places where it says $g(x)$ in the equation? What is meant by the "around $x=1$" part; isn't this somewhat redundant information?
If the above statements are true, then I'm just isolating $g(x)$, taking the derivative, and plugging in $\frac{\pi}{2}$ as $g(x)$, correct?