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I really want to know how can I check if my derivative is correct without using a calculator just to check in the exam

For ex: $x^2 + y^2 = 0$

I know that $\frac{dy}{dx} = -\frac{x}{y}$

But how can I check if it's correct in the exam if harder problems were given?

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    $\begingroup$ $x^2+y^2=0$ is a bad example! It is a single point. $\endgroup$ – Emilio Novati Jun 9 '17 at 20:42
  • $\begingroup$ In the case that you could solve for $y$ as a function of $x$, you could see whether implicit differentiation and explicit differentiation give the same result. $\endgroup$ – Foobaz John Jun 9 '17 at 21:34
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I think the only way to check would be to integrate the derivative and see if you get the original function back; I don't know of any other way to check without a calculator.

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Heuristically, you can draw the shape that is described (in this case, the circle), draw the tangent line to it, and see if various properties (ex., if the slope is negative or positive, if it's large or if it's small, etc.) match the properties of the curve you've drawn. Of course, this has the problem that figuring out what the shape should be, if you're not already familiar, can (and will) eat up valuable exam time.

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