Posted by Joel on October 31, 2002 at 19:12:38:
In Reply to: Re: Implicit Differentiation posted by Jack on October 31, 2002 at 10:49:46:
: : : I find these kind of problems easier to do if Implicit Differentiation is used.
: : Please give an example
: :#2 V=4pi/3r^3
: : dV/dt = 4pir^2 dr/dt = 8pir^2
Thanks Jack. I hate to be picky, but that's not implicit differentiation. That's the chain rule. I don't know how to precisely define implicit differentiation, but an example would be to find dy/dx from an equation like this:
x^2 + 4y^2 = 25
without first solving for y. As in:
2x + 8y*dy/dx = 0
8y * dy/dx = -2x
dy/dx = -x/(4y)
At this point you might proceed to solve the original for y & substitute to get an expression for the derivative that doesn't have a "y" in it. Sometimes, depending on the specific problem, you don't need to do that, and sometimes you can't.
Jack, on a completely different subject, would you please see what your mathcad will do with the following. I had to integrate sqrt(x^2 + 1/2 + 1/(16x^2)). That square root equals (x + 1/(4x)) and so the integral I got by hand is (x^2)/2 + (ln(x))/4 .
Then I tried to check my answer using "The Integrator" on Wolfram's website. If I ask it to integrate (x + 1/(4x)) it gives the same answer I got. But if I ask it to integrate EITHER sqrt(x^2 + 1/2 + 1/(16x^2)) or sqrt((x + 1/(4x))^2) the answer it gives is sqrt(-1/(16x) + x/2 + (x^3)/3). I don't see how those two answers can be equivalent (although I guess they might be). What does mathcad do with these integrals?
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