Can you use derivatives and integrals to solve algebraic problems?

I've seen derivatives and integrals used before to change the form of an equation to one that is more suitable for solving a problem. Usually the person will just differentiate both sides and the equality holds? Is this always the case?

Because if you take the derivative of both sides of x^2 = 4, you get 2x = 0, which is not right. Can someone explain how this works?


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I am not aware that you can do such things, do you have an example of where this was used?

But yes the equality will hold once you do the same thing on both sides of the equation.
What you have found in your example is the value of x for which x^2 - 4 has a horizontal tangent.

What you might be thinking of is that if two functions are identically equal (equal for all values of x), then their deriviatives are equal. In your example, the functions x^2 and 4 (or x^2 - 4 and 0) are not identically equal; they are only conditionally equal.
In my book they do use the derivative to solve an infinite sum. [Broken]

[edit] Ah ok! Thanks!
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This summation is identically true for all values of alpha, so the derivatives will also be equal.
[tex]\sum_{i = m}^n \alpha^i~=~\frac{\alpha^m - \alpha^{n + 1}}{1 - \alpha}[/tex]

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