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I know in general you cannot take the derivative of an equation to solve it because the derivative at a point depends on neighboring points of a function.

However, lots of the proofs done in my probability course, for example finding the variance of a geometric random variable is done by differentiating both sides. Why is this allowed?

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    $\begingroup$ A sufficient condition, if you're differentiating with respect to $x$, is that both sides of the equation should be equal for all values of $x$ in some interval. In that case, the left- and right-hand sides are different ways of writing the same function of $x$. In general, if the equation is only true for one value of $x$, then you can't do it. $\endgroup$ – user49640 Mar 30 '17 at 6:38
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considering the equation $$x^3 = x + 6$$ here $x=2$ is the solution of differential equation but if we differentiate both sides of equation then we get $$3x^2 = 1$$ in this case $x=2$ gives $12=1$ not satisfied this means that two functions are just interesect not tangent to each other If these function are actually equal then we can differenctiate both sides For more geometric perspective the arithematic operation act pointwise but the drivative requires the knowledge of nieghborhood This means that derivative is not operation function if you sketch $x^3$ and $x+6$ then you would get answer of your each question....

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