$1.$ If $\backslash ($ V $\backslash )$ is the volume of a cube with edge length $\backslash ($ x $\backslash )$ and the cube expands as time passes, find $\backslash ($ d V $/$ d t $\backslash )$ in terms of $\backslash ($ d x $/$ d t $\backslash ).$

$2.($a$)$ If $\backslash ($ A $\backslash )$ is the area of a circle with radius $\backslash ($ r $\backslash )$ and the circle expands as time passes, find $\backslash ($ d A $/$ d t $\backslash )$ in terms of $\backslash ($ d r $/$ d t $\backslash ).$

$($b$)$ Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of $\backslash (1\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ how fast is the area of the spill increasing when the radius is $30$ m $?$

$3.$ Each side of a square is increasing at a rate of $\backslash (6\backslash$mathrm$\{$~cm$\}/\backslash$mathrm$\{$s$\}\backslash ).$ At what rate is the area of the square increasing when the area of the square is $\backslash (16\backslash$mathrm$\{$~cm$\}^\{2\}\backslash )?$

$4.$ The length of a rectangle is increasing at a rate of $\backslash (8\backslash$mathrm$\{$~cm$\}/\backslash$mathrm$\{$s$\}\backslash )$ and its width is increasing at a rate of $\backslash (3\backslash$mathrm$\{$~cm$\}/\backslash$mathrm$\{$s$\}\backslash ).$ When the length is $20$ cm and the width is $10$ cm $,$ how fast is the area of the rectangle increasing?

$5.$ A cylindrical tank with radius $5$ m is being filled with water at a rate of $\backslash (3\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$min$\}\backslash ).$ How fast is the height of the water increasing?

$6.$ The radius of a sphere is increasing at a rate of $\backslash (4\backslash$mathrm$\{$~mm$\}/\backslash$mathrm$\{$s$\}\backslash ).$ How fast is the volume increasing when the diameter is $80$ mm $?$