Implicit Differentiation and Related Rates

Given \(3x^2+\sqrt[4]{y}=19\), find \(\dfrac{dy}{dx}\) using implicit differentiation.

Given \(x^2y^9-e^y=6-4x^3\), find \(\dfrac{dy}{dx}\) using implicit differentiation.

An expandable sphere is filled with liquid at a constant rate from a tap (imagine a water balloon connected to a faucet). The radius of the sphere is increasing at a rate of 2 inches per minute when the radius is 3 inches. How fast is the liquid flowing from the tap (i.e., how fast is the volume of the sphere changing) when the radius is 3 inches? Round you answer to three decimal places, if necessary.

A company that manufactures and sells high-end calculators has a weekly revenue equation given by \(R=-0.01x^2+400x\) dollars, where \(x\) is the number of calculators sold each week and a weekly cost equation given by \(C=100x+10,000\) dollars, where \(x\) is the number of calculators produced each week. If the number of calculators produced and sold is continuing to increase at a rate of 200 calculators per week, find the rate of change of profit with respect to time when 6000 calculators are produced and sold each week. Interpret your answer.

A 25-foot ladder is leaning against a wall. If the bottom of the ladder is pushed toward the wall at a rate of 1 foot per second and the bottom of the ladder is initially 20 feet away from the wall, how fast does the top of the ladder move up the wall after 5 seconds of pushing? Round your answer to two decimal places, if necessary.