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So I was told to find the rate of change of

$$ f(x,y) = x^2 − 3xy + y2 $$ with respect to $$r(t) = e^{2t}+t^2$$ I know usually I would take the derivative with respect to each variable and then plug in a point which is usually given (finding the gradient), but what do I do if I'm given another equation?

Question

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  • $\begingroup$ As written, this doesn't seem to make much sense. Could you post the full text of the problem? $\endgroup$ – Cameron Buie Apr 13 '15 at 18:08
  • $\begingroup$ @CameronBuie I added a picture of the question, hopefully if it helps with the explanation $\endgroup$ – Joseph hooper Apr 13 '15 at 18:33
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You want the rate of change of $f(x,y)$ with respect to $t$ along the curve $$\vec r(t)=e^{2t}\hat i+t^2\hat j,$$ using the chain rule.

First of all, then, apply the chain rule to find $\frac{df}{dt}$ in terms of partial derivatives and direct derivatives. Once you've done that, you'll substitute $x=e^{2t}$ and $y=t^2$ into the formula you've derived via the chain rule.

An alternate approach that you can use to check your work is to simply take the derivative of $f\left(e^{2t},t^2\right)$ with respect to $t$.

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