How do I use specifically the chain rule for $x^{\sin(x)}$?
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$\begingroup$The power rule of differentiation $$ (x^u)'=ux^{u-1} $$ is only valid for a constant $u\in\mathbb{R}$. It is not valid if $u$ is a function of $x$ itself.
$\endgroup$ 1 $\begingroup$In your application of the chain rule you have defined
$$ v(u) = x^u, \quad \text{and} \quad u(x)=\sin(x), $$
and noticed that it follows that
$$ v(u(x)) = x^{\sin(x)}, $$
in order to apply the rule to $v(u(x))$.
The problem is that $v(u)$ as you defined it also depends on $x$, which is the argument of the "inside function", $u(x)$. This invalidates the application of the rule.
$\endgroup$ 2 $\begingroup$Treating the exponent as a constant, the derivative is $\sin(x)\cdot x^{\sin(x)-1}$, treating the basis as a constant the derivative is $x^{\sin(x)}\cdot\ln(x)\cdot\cos(x)$. Adding both gives the derivative of $x^{\sin(x)}$. (This is not a coincidence.)
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