29. - / 1 points find the derivative of the function. h(x) = (arccos(x)) ln(x) h(x) =

Explanation:

Step1: Apply product - rule

The product - rule states that if $h(x)=u(x)v(x)$, then $h'(x)=u'(x)v(x)+u(x)v'(x)$. Here, $u(x)=\arccos(x)$ and $v(x)=\ln(x)$.

Step2: Find derivative of $u(x)$

The derivative of $u(x)=\arccos(x)$ is $u'(x)=-\frac{1}{\sqrt{1 - x^{2}}}$ for $|x|\lt1$.

Step3: Find derivative of $v(x)$

The derivative of $v(x)=\ln(x)$ is $v'(x)=\frac{1}{x}$ for $x\gt0$.

Step4: Substitute into product - rule

$h'(x)=u'(x)v(x)+u(x)v'(x)=-\frac{\ln(x)}{\sqrt{1 - x^{2}}}+\frac{\arccos(x)}{x}$.

Answer:

$-\frac{\ln(x)}{\sqrt{1 - x^{2}}}+\frac{\arccos(x)}{x}$