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I have a question. How in general would one differentiate a composite function like $F(x,y,z)=2x^2-yz+xz^2$ where $x=2\sin t$ , $y=t^2-t+1$ , and $z = 3e^{-1}$ ? I want to find the value of $\frac{dF}{dt}$ evaluated at $t=0$ and I don't know how. Can someone please walk me through this?

I tried a couple of things, including chain rules and jacobians. I know that $\frac{dF}{dt}$ should equal $\frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt} + \frac{\partial F}{\partial z} \frac{dz}{dt}$ but for some reason this doesn't work, or I am doing something wrong. I start out by differentiating to get $\frac{\partial F}{\partial x}=4x+z^2$, $\frac{\partial F}{\partial y}= -z$, $\frac{\partial F}{\partial z} = 2xz-y$, $\frac{dz}{dt}=0$, $\frac{dx}{dt}=2\cos t$, $\frac{dy}{dt}=2t-1$ but this doesn't match the answer, which my book says is $24$.

How do they get this, and where is my error? Thanks.

Update:

What I get is as follows: $F(x,y,z)=2x^2-yz+xz^2$, $\frac{\partial F}{\partial t}=\frac{\partial F}{\partial x} \frac{dx}{dt} + \frac{\partial F}{\partial y} \frac{dy}{dt} + \frac{\partial F}{\partial z} \frac{dz}{dt}$,$\frac{\partial F}{\partial t}=(4x+z^2)(2cos(t))-z(2t-1)$ Which for $t=0$ gives $x=0$ and $\left. \frac{\partial F}{\partial t} \right|_{t=0} = 2z^2+z=9e^{-2}+3e^{-1}$ which clearly isn't $24$ so I must be doing something completely wrong.

Edit: I want to rephrase the question. Since everyone else I have talked to thinks there was an error in the book, does everyone here agree?

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    @DonAntonio If you take $z = 3 e^{-t}$ you get the desired result. There must be a typo on the book :)2012-10-24

2 Answers 2

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$\frac{\partial F}{\partial t}=\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial y}\frac{dy}{dt}+\frac{\partial F}{\partial z}\frac{dz}{dt}=$

$=(4x+z^2)\cdot 2\cos t-z(2t-1)+(2xz-y)\cdot 0$

You'll now to substitute:

$t=0\Longrightarrow\,x=0\,,\,y=1\,,\,z=3\,e^{-1}$

The final result is, if I'm not wrong,

$9\,e^{-2}+3\,e^{-1}$

which has nothing to do with $\,24\,$, so either the book (which one, btw?) has a mistake or you miscopied the exercise.

Ps. Please do you check that according to what you wrote $\,z\,$ is independent of $\,t\,$...

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You need to use implicit differentiation as one of your tags suggests.

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    You are missing $\partial F/\partial t$ in your formula for the total derivative.2012-10-24