Differentiate the following making $\frac{dy}{dx}$ the subject.

$($a$)$

$2xy=y^3+x^2-x$

$($b$)$

$logy=6^{6x}$

$($c$)$

$y=2\sqrt[\phantom{2}]{x+2y}$

The radius of a circle is increasing at a rate of $12c\frac{m}{s}.$ How fast is the area increasing at the instant

when the radius has reached $7cm?$

A $15-$foot ladder when it starts to slide down the wall at a rate of $2\mathrm{.}5$ feet per minute. How fast is the

base of the ladder moving the height of the ladder is $12$ feet from the ground at that instant?

The cost to manufacture $x$ cell phones in a day is

$C(x)=12000+15x+\frac{x^2}{10000}$

The daily production level is currently $x=4000$ cell phones and is increasing at a rate of $125$ units

per day. How fast is the average cost changing?