Math Is Fun Forum

Taylor S

Guest

Derivatives in form of f(x) and g(x)

Let f and g be differentiable functions. Find d/dx [f((g(xf(x)))^2)] in terms of f, g, f', and g'.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Derivatives in form of f(x) and g(x)

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)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.