how to use implicit differentiation

[kristoball]

hello, how would one go about differentiating V=W*L*H using implicit differentiation? I can get as far as V`=(W*H*L)` so any and all help would be appreciated.

Thanks!

 

[tkhunny]

You should decide what is implied in the definition.

Is W a function of H?
Is H a function of L?
Is L a functio of W?

 

[kristoball]

its the formula for the volume of a cube where V = volume, W = width, L = length, H = height. I should have mentioned that. Sorry. The entire question is as follows (if it helps)

A collapsing room (with walls and ceilings both moving towads the center of the room) has a length that is shrinking by 8 ft/min, a width shrinking by 2 ft/min, and a height decreasing by 5 ft/min. What is the rate of change of volume when the dimensions of the room (W, L, H) are 9, 14, and 9 feet, respectively.

 

[tkhunny]

There you go. Everything is a function of time. You will need the Product Rule along with your Implicit Chain Rule.

V(t) = H(t) * W(t) * L(t) = H(t) * [W(t) * L(t)] - With a little grouping to give you a hint.

 

[kristoball]

thanks, that certainly helps. how do i go about doing implicit differentiation...?

thanks!

 

[tkhunny]

That's why I grouped the last two, so you might see it.

Can you do a right circular cyllinder?

\(\displaystyle V(t) = \pi\cdot r(t)^{2}\cdot h(t)\)

Or, how about a sphere?

\(\displaystyle V(t) = \frac{4}{3}\cdot\pi\cdot r(t)^{3}\)

Here's the Sphere:

\(\displaystyle \frac{dV}{dt} = 4\cdot\pi\cdot r^{2}\cdot \frac{dr}{dt}\)

 

[soroban]

Hello, kristoball!

This does not involve implicit differentiation.
And it certainly helps when you give us the entire problem . . .

A collapsing room (with walls and ceilings both moving towads the center of the room)
has a length that is shrinking by 8 ft/min, a width shrinking by 2 ft/min, and a height decreasing by 5 ft/min.
What is the rate of change of volume when the dimensions of the room (L, W, H) are 14, 9, and 9 feet, respectively?

We have: .\(\displaystyle V \;=\;LWH\)


Differentiate with respect to time:

. . \(\displaystyle \displaystyle \frac{dV}{dt} \;=\;LW\frac{dH}{dt} + LH\frac{dW}{dt} + WH\frac{dL}{dt}\) .[1]


\(\displaystyle \text{We are given: }\:\begin{Bmatrix}L = 14 && \frac{dL}{dt} = -8 \\ W = 9 && \frac{dW}{dt} = -2 \\ H = 9 && \frac{dH}{dt} = -5 \end{Bmatrix}\)


Substitute into [1].

Got it?