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 $t=\sin (t+z)$.

Determine the coordinates of the points where the curve $x^2-2xy+4y^2=8$ intersects the $x$-axis, and find the slopes of the curve at these points.

Consider the parabola $x=-2y^2$.

From the point $(c,0)$, it is possible to draw three normal lines to the curve. One of these normal lines is the $x$-axis. Determine the value of $c$ so that the other two normal lines are perpendicular to each other.