# Example 2

• 3.9.12.
• A particle is moving along a hyperbola $$xy=8$$. As it reaches the point $$(4,2)$$, the $$y$$-coordinate is decreasing at a rate of 3 cm/s. How fast is the $$x$$-coordinate of the point changing at that instant?
• Here, the expression relating $$x$$ and $$y$$ is already given to us, so we'll have to derivate it implicitly.
• What you need to look out for is that the variable with respect to which we're differentiating is $$t$$, the time.
• Therefore implicit differentiation yields the following. $x'y+xy'=0$
• Substituting the known numerical values, we get the following. $x'\cdot2+4\cdot(-3)=0.$
• This yields $$x'=6$$ cm/s.

# Example 3

• 3.9.14.
• If a snowball melts so that its surface area decreases at a rate of $$1\,\mathrm{cm}^2/\mathrm{min}$$, find the rate at which the diameter decreases when the diameter is 10 cm.
• The area of a sphere is $$A=4r^2\pi$$. But we need an equation with the area $$A$$, and the diameter $$d=2r$$.
• Therefore, we want to use $A=4(d/2)^2\pi=d^2\pi.$
• Implicit differentiation gives $A'=2dd'\pi$
• Substituting the given numerical values, we get $-1=2\cdot10\cdot d'\cdot\pi$
• Therefore, we get $$d'=-\frac{1}{20\pi}$$ cm/min.
• Exercises. 3.9: 4, 6, 8, 16, 18, 22, 24, 30, 38