Use implicit differentiation to calculate $\frac{dy}{dx}$ for the equation $x^2+y^2=25$.

Determine the equation of the normal line to the following curve at the given point:

$x^3+y^3=8xy$; $(4,4)$

Graph the tangent line and normal line to the following curve at the given point:

$2x^4=81(x^2+y^2)$; $(9,9)$

Evaluate and simplify $z^{\prime }$ given $t=\sin (t+z)$.