= Tangent stiffness
{title2=$k_{\rm tan}=dF/dx$}
The \b[tangent stiffness] at a point on a differentiable <force-displacement curve> is its local slope,
$$
k_{\rm tan}(x_0)=\left.\frac{dF}{dx}\right|_{x_0}.
$$
It governs the first-order response to a small <displacement> change:
$$
F(x_0+\Delta x)-F(x_0)=k_{\rm tan}(x_0)\Delta x+o(\Delta x).
$$
The graphical interpretation is the slope of the <tangent line>. Structural <stiffness> measured from force-deflection behaviour includes geometry as well as material response.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Discussion following the axial force-displacement relation]{ref}
For the illustrative model $F(x)=ax+bx^3$, choose $a=100\,\mathrm{N\,m^{-1}}$ and $b=1000\,\mathrm{N\,m^{-3}}$. At $x_0=0.10\,\mathrm m$, the <tangent stiffness> is $130\,\mathrm{N\,m^{-1}}$. An additional $\Delta x=0.001\,\mathrm m$ therefore predicts a force increase of $0.130\,\mathrm N$ to first order. Direct substitution gives $0.130301\,\mathrm N$; the difference is the higher-order contribution omitted by the tangent approximation. The <secant stiffness> from the origin is only $110\,\mathrm{N\,m^{-1}}$ at this point.
In a displacement-controlled experiment, one can estimate the local slope with a central <finite difference>,
$$
k_{\rm tan}(x_0)\approx
\frac{F(x_0+h)-F(x_0-h)}{2h}.
$$
Making $h$ smaller reduces the approximation error for a smooth exact curve, but measured differences can become dominated by <measurement uncertainty>. The force channel, displacement channel and loading branch must therefore be specified. Crosshead movement can include machine <mechanical compliance>, while an <extensometer> records a selected specimen extension.<Extensometers in the Instron documentation>[When to use an extensometer: machine compliance, grip seating and load-train deflection]{ref}
For a curve with <hysteresis>, loading and unloading can have different local slopes at the same displacement. At a sharp corner a unique <derivative> may not exist, and a finite-interval slope should be reported as such.
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