Question

If g(x) + xsin(g(x)) = x^5 and g(4) = 0 , find g '(4)

Answer 1

\[g(x)+x\sin(g(x))=x^5\]
Differentiate both sides and solve for \(g'(x)\), you have
\[\begin{align*}g'(x)+\sin(g(x))-x\cos(g(x))~g'(x)&=5x^4\\\\
g'(x)&=\frac{5x^4-\sin(g(x))}{1-x\cos(g(x))}
\end{align*}\]
Plug in \(x=4\) and make use of the given data points.