Calculus (First Year) - Derivatives Question 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?

I do not think you will succeed in isolating $g(x)$. (For sure, I would not be able to.)

Just differentiate immediately. We get $g'(x)+x^3g'(x)\cos(g(x))+3x^2\sin(g(x))=4x^3+4$.

Finally, put $x=1$, and solve for $g'(1)$.

You will need $\cos(g(1))$ and $\sin(g(1))$, but these are available from the given information.

**Remark:** One cannot find $g'(1)$ just from the information we have **at** $x=1$. We need to know that the given equation holds in a neighbourhood of $x=1$.