this is an implicit differentiation problem

(note that we used the product rule for the middle term on the left hand side)

now the rest is algebra. solve for

we have , we proceed by the chain rule2. Differentiate

note that the last factor is the derivative of by the quotient rule

the volume, , of a sphere with radius is given by3. A spherical balloon is being inflated. If the volume of the balloon is increasing at a rate of , how fast is the radius increasing when the radius is 3m?

differentiating implicitly with respect to time, we have

we are told and we want , so plug these in and solve for

yes, draw a diagram. note that it makes a right triangle, the hypotenuse is the length of the rope from the pulley to the boat, call this length , the base is the distance of the boat from the dock, call this distance . The height is 1. Now we are told that (note that it is negative because the distance is decreasing) and we want when4. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the boat is pulled in at a rate of 0.8 m/s, how fast does the boat approach the dock when it is 10m from the dock?

By Pythagoras' Theorem, we have that

............................(1)

now plug in everything on the right to find . note that you can find the value of from equation (1)