Which quantity gives the resistance of a component?

In a current vs potential difference graph, we can obtain the value of the resistance of the component. There are books that say gradient-inverse is the resistance and also books that say the value of $\frac{\text{current, I}}{\text{potential difference:V}}$, which comes from the coordinates of the graph, provides the resistance.

I tend to believe the second method (coordinates) provide the true value of resistance. The reason is because when we discuss gradient, we are talking about change wrt to another quantity. e.g. acceleration is change of velocity wrt time. Resistance is NOT change of potential difference wrt current. Resistance is the ratio of potential difference over current, according to Ohm's law.

This is frustrating because I can find official solutions to international certified examinations also saying that the gradient method is the correct one. So can someone who is very knowledgeable in this area provides the true answer? Students need to be taught the correct methods.

Update:

The two ways to calculate the resistance from an I-V graph is:

1. Take the inverse of the gradient. In the diagram ,it would be $\frac{10-1}{15-5}=\frac{9}{10} \Omega$.
2. Take the ratio of $\frac{V}{I}$ which in this case will be $\frac{10}{5} = 2 \Omega$.

These two methods yield different results, but both give the same dimension. Which one is the correct method?

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Because there are lots of variables having some impact on resistivity (like temperature), in non-linear resistive materials the resistivity can be also a function of provided current or voltage, there is need to have true function combining voltage drop and current value on the material, like I = f(U). Because sometimes this function is strongly non-linear (for semi-conductor materials like eg. a diode), it's easier to represent this dependency (some mathematical function in fact) on a IU graph. In this case the resistivity (or resistance) IS a derivative (or gradient if you wish) of voltage drop vs current (or vice-versa).

When you measure both voltage and current and calculate resistance you usually do this having constant input(s). So you don't see then this derivative connection, but -- relating to your example -- if you move with constant speed you don't see that the path changes constantly with the time. I think you misunderstand acceleration (velocity change in time) as analogue of resistance (voltage change in current). I think more proper would be to imagine that resistance is your speed: you increase the input voltage and with this speed current changes.

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I am not talking about input voltage (emf). I am talking about potential difference in a load, such as a resistor. Ohm's law states that the current through a resistor is directly proportional to the p.d. across. If I talk about a general component that do not obey Ohm's law, then resistance is just the value obtained when I divide the p.d. by the current across (by doing an experiment). Resistance is empirical. So how can resistance be calculated by taking gradient since it is not even change in voltage over current? –  Standstill Jun 4 '13 at 7:54
Why need you think it is voltage change? Think about current change because of voltage input. Take a look at your graph; U is in x-axis. –  Voitcus Jun 4 '13 at 8:33
Then the resistance would be $2 \Omega$ (in the graph) if I understood you correctly? –  Standstill Jun 4 '13 at 9:18
In this specific point P, yes. Mathematically, you should consider derivative, and the angle of the orange line is tangent from U/I = 2, which leads to dI/dU = 0.5 = 1/R. But I'm trying to explain other way: if you increase voltage by dU, the current will increase by dI, and dI = dU / R. In typical (simplest) Ohm's law, the R is constant, so this is the same for all Us. In the graph presented it is non-linear function, so small (infinitely) changes should be considered. –  Voitcus Jun 4 '13 at 9:29

The rate of change of voltage with respect to current is known as the dynamic resistance. The ratio of voltage to current at a point is the static resistance.

For an ohmic circuit element, the static resistance and the dynamic resistance are equal.

For non-linear circuit elements, the dynamic resistance is more useful as one can then linearize the element about an operating point and, for small variations about the operating point, the element behaves as a resistor with resistance equal to the dynamic resistance.

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Ohm's Law is indeed V = IR, which means the resistance is R = V/I.

This article defines resistance of an ideal conductor as having the potential difference being proportional to the current through it.