Question

What does the gradient of a stress-strain graph tell you?

Answer 1

A stress-strain graph represents a dependence between axial normal stress $\sigma$ and axial normal strain $\varepsilon$ of materials. In the linear region (where the stress obeys Hook's law):

$$\sigma = E\varepsilon$$

The slope here is:

$$Slope = \sigma' = E$$

which is Young's modulus.

In general, when $\sigma = f(\varepsilon)$, the slppe represent the instant rate of change of stress when strain changes.