Related Rates - Volume of cylinder

[JimmysJohnson]

G'day again.
I was wondering if i could get some help on my understanding of this question.
"A cylinders radius is decreasing at the rate of 2cm/sec while it height is increasing at a rate of 3cm/sec. How fast is the cylinders volume changing when its radius is 10cm and its height 18cm?"

Since we know V = (pi)r²h
I thought that we could just use implicit differentiation to find the rate of change in respect to time. But in doing so i imagined that we'd simply use the product rule for the three terms (im not sure if this is where the flaw happens, but after a google search it seems as though it is possible).
After which we'll get
dV/dt = (r²h)+((pi)(2r)(dr/dt)(h))+((pi)(r²)(dh/dt))
However when i sub in the respective points to solve for the rate of change of volume, i get very far off from the actual equation.
I understand that in order to solve the equation properly we take pi out of the equation
pi(r²h) and to go from there. I'm just not sure why my method doesn't work since its probably the methodology i'd instantly go for. Is it because pi is a constant and as such we can't differentiate it?
Thank you for your time and any/all help.

 

[MarkFL]

I would begin with:

[MATH]V=\pi r^2h[/MATH]
And now differentiate with respect to time \(t\):

[MATH]\frac{dV}{dt}=\pi\left(2rh\frac{dr}{dt}+r^2\frac{dh}{dt}\right)[/MATH]
And now plug in the given data. If you try to include the constant \(\pi\) as a function to be differentiated, you must use:

[MATH]\frac{d}{dt}(\pi)=0[/MATH]

 

[HallsofIvy]

You have "dV/dt = (r2h)+((pi)(2r)(dr/dt)(h))+((pi)(r2)(dh/dt))"
(It would be better to write "r^2" rather than "r2" to make the exponent clearer.)
Where did the "r^2h" come from? Was that the derivative of \(\displaystyle \pi\)? If so why isn't there a "\(\displaystyle (d\pi/dt)\)"?

You ask "Is it because pi is a constant and as such we can't differentiate it?"
Of course we can differentiate a constant! One of the first things you learned in Calculus I is that the derivative of any constant is 0!