First derivative on 2 variables

Question

I got a problem that is causing me a headache. So differentiating $\frac {1}{12π}*c^2*h$.

According to the constant rule we can pull out the constant $\frac {1}{12π}$ and differenciate $c^2*h$ as two different variables using the chain rule along with the product rule? The problem here is that my solutions doesn't match the online calculators which are taking $c^2$ as a constant and therefore produce the following answer $c \cdot \frac {h}{6π}$. I don't understant what I'm I doing wrong

With respect to time (t) so V=volume, (d/d(t)V= d/d(t)(1/12π)c^2h — David, Nov 18 '18 at 00:09
in that case $c$ or $h$ (at least one of them) must be a function of $t$ or else the answer is $0$. — valer, Nov 18 '18 at 00:13
If you prob write c(t) instead of c, then the online calculators should work (and of course h(t) instead of h) — Stan Tendijck, Nov 18 '18 at 00:54

score 1 · Answer 1 · answered Nov 18 '18 at 03:19

1

In the most general case, both $c$ and $h$ are functions of $t$. Then you can write:

$\frac{dV(t)}{dt}$

= $ \frac{1}{12 \pi}\frac{d}{dt}c^2(t)h(t)$

= $\frac{1}{12 \pi} \left( 2c(t)c'(t)h(t) + c^2(t)h'(t)\right)$.

That's as far as one can get until you tell us how $c$ and $h$ vary as functions of $t$.

answered Nov 18 '18 at 03:19

Godfather

2,355

Hi thanks for your answer. So C is the circumference of a tree and H is the height. I need to find the rate of grow or the rate of change of the volume of the tree. As you can see (1/12π)c^2h is the formula for the volume of the tree. You're right but at the time of applying the first derivative using the product rule and pulling out the constant 1/12π I left with "c^2+h/6π" as an answer. When checking my result with online calculators the gave me a different answer specifically ch/6π – David Nov 18 '18 at 21:08

First derivative on 2 variables

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