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As a diode doesn't follow $V$ proportional to $I$, I was wondering if we can use Ohm's law to calculate its resistance for a specific set of voltage and current, as its resistance varies as per the voltage applied.

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Yes, resistance is $R=V/I$, even if $V/I$ is not constant. You would just have something like $R=R(I)=V(I)/I$, where $R(I)$ and $V(I)$ are functions of $I$ (or you could do a similar expression with functions of $V$).

To be nitpicky, $V=IR$ is not Ohm's law. Ohm's law is that $R=V/I$ is constant. So by using $R=V/I$ you are just using the definition of resistance. Usually in introductory physics you don't consider cases where Ohm's law doesn't apply, so Ohm's law and $V=IR$ become synonymous, but when digging deeper one should be aware of this.

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Well semiconductors behave quite different than conductors, the I-V characteristics aren't linear like conductors and in fact, they even differ when they are forward biased and reverse biased.

So to avoid all these, instead of resistance, we define dynamic Resistance which is given as $$R_{dynamic}=\frac{\Delta V}{\Delta I}$$

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