$\backslash$table$[[x,-1,0,1,2],[g(x),6,4,2,-1],[g^{\text{'}}(x),-1,-7,-2,-3]]$

The table above gives selected values for a differentiable and decreasing function $g$ and its derivative. Let $f$ be the function with $f(1)=0$ and derivative given by $f^{\text{'}}(x)=xsin(x^2).$

$($a$)$ Find $f^{\text{'}\text{'}}(e^{1\mathrm{.}5}).$ Express your answer as a decimal approximation.

On your AP Exam, you will handwrite your responses to free$-$response questions in a test booklet

$($b$)$ Let $h$ be the function defined by $h(x)=f(g(3x)).$ Find $h^{\text{'}}(0).$ Express your answer as a decimal approximation.

$$ C$)$ write an equation for the line tangent to the graph of g$^-1,$ the inverse function of g$,$ at x$=-1.$

$$ D$)$ The point $(1,4)$ lies on the curve in the xy$-$plane given by the equation f$($x$)+$g$($x$))^2=$xy$.$ What is the value of dy$/$dx at the point $(1,4)?$ Express your answer as a decimal approximation.