Product rule

[JohnGoodwill]

Hello
I'm dealing with a problem I can't wrap my head around. It's just simple product rule, but I don't get the right result.

I am given:
[MATH] V = H \cdot L \qquad, H = a \cdot \sqrt{L} [/MATH]and I want to calculate the change of V with respect to time [MATH] \frac{dV}{dt}[/MATH].
Writing it like this, using product rule:

[MATH] \frac{dV}{dt} = \frac{d}{dt}(H \cdot L) = H \frac{dL}{dt} + L \frac{dH}{dt} [/MATH]
Now, I get

[MATH] \frac{dV}{dt} = a \cdot \sqrt{L} \cdot \frac{dL}{dt} [/MATH]
I want an expression in terms on the change in L with respect to time, hence why I keep it.
However the right result is:

[MATH] \frac{dV}{dt} = \frac{3}{2}a \sqrt{L} \frac{dL}{dt} [/MATH]and I don't understand why...
The H term is being differentiated, it seems.

 

[Dr.Peterson]

How did you drop the second term in the product rule? Are you assuming H is a constant, so dH/dt = 0? Clearly it isn't.

 

[JohnGoodwill]

How did you drop the second term in the product rule? Are you assuming H is a constant, so dH/dt = 0? Clearly it isn't.

Yes, I did assume it to be constant, but you're right, it isn't.

 

[Steven G]

Why use the product rule at all? Why not multiply a*sqrt(L) and L, get a*L^3/2 and take the derivative of that?

 

[Steven G]

Yes, I did assume it to be constant, but you're right, it isn't.

If you thought that H was a constant then you should have said that the dV/dt = H*dL/dt = .... That is why use the product rule at all, just use the constant multiple rule.

 

[JohnGoodwill]

If you thought that H was a constant then you should have said that the dV/dt = H*dL/dt = .... That is why use the product rule at all, just use the constant multiple rule.

Yes, you're absolutely right. No need to use the product rule.
Thank you both for your responses!