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Let f and g be differentiable functions. Find d/dx [f((g(xf(x)))^2)] in terms of f, g, f', and g'.
Chain rule madness!
Not really worth LaTexing:
f((g(xf(x)))²) = f(h(x))
h(x) = g(xf(X)))² = k(x)²
k(x) = g(xf(X)) = g(m(x))
m(x) = xf(x)
Now, Working backwards I'm going to take the derivative of each function I defined:
m'(x) = xf'(x) + f(x) product rule
k'(x) = g'(m(x))m'(x) = g'(xf(x))(xf'(x) + f(x)) chain rule
h'(x) = 2k(x)k'(x) = 2g(xf(x))(g'(xf(x)))(g'(xf(x))(xf'(x) + f(x))) product rule
SO:
d/dx f(h(x)) = f'(h(x))h'(x) = f'((xf(X)))²)(2g(xf(x))(g'(xf(x)))(g'(xf(x))(xf'(x) + f(x))))
Well, that was a pain. With problems like these it's easier (for me at least) to color code them. Maybe I'll do that tonight, I have to go now.
PS I hope everyone looks at my work, THERE'S A LOT OF ROOM FOR ME TO MAKE A STUPID MISTAKE.
Last edited by bossk171 (2007-09-24 04:50:23)
There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.
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