When is it justified to differentiate both sides of an equation? I'm just reading a textbook, which out of the blue takes the derivative of both sides, without any real justification. For instance, I can't take the derivative of , and still preserve that equality.
Thanks in advance,
If two functions f(x) and g(x) are identically equal then you can differentiate both sides of the equation f(x) = g(x), and the resulting equation f'(x) = g'(x) will also be identically true. For example, (for all x). Differentiate both sides and you get , also true for all x.
But the equation is not an identity. It is only true for two values of x. When you differentiate it you get a completely different equation, which has a different interpretation, as skeeter has pointed out.