Equality and The Derivative

[terryphi]

I know this is a simple question, but if I take the derivative of the left hand side, do I have to take the derivative of the right hand side? How do I maintain equality with the derivative? it's not an algebraic operation, so I'm not sure if I can just do the same thing to both sides and it still be equal.

what about integration? for integration, if I have to take the integral of both sides, how do I pick my bounds?

 

[HallsofIvy]

General rule: if A= B and you do the same thing, say "f", to both sides, then f(A)= f(B). As long as "doing something" to A changes it, then A= B cannot give f(A)= B. That is true of any operation, whether algebraic, differential, or integral. As for "how do I pick my bounds", that depends on the particular problem. If A= B then \(\displaystyle \int Adx= \int B dx\) for any bounds or even as a indefinite integral.

 

[TylerH]

The derivative, per se, isn't an algebraic operator, but differentiation is. The dy/dx is just moving the dx from the right side.
$$y=x^2$$
$$d(y)=d(x^2)$$ Apply the operator of differentiation, "d", to both sides.
$$dy=2x\:dx$$ dy is just dy, and d(x^2) is 2x dx

To get it into the familiar derivative form, just divide by dx
$$\frac{dy}{dx}=2x$$

For integration in equalities, just do the indefinite integral, aka anti-derivative, of both sides.