Tangent Line at x₀ on y = x³ − 3x

At $x_0$, the tangent has slope $f'(x_0) = 3x_0^2 - 3$ and passes through $(x_0, x_0^3 - 3x_0)$. The equation is $y - f(x_0) = f'(x_0)(x - x_0)$. Local max and min occur where $f'(x_0) = 0$, i.e. $x_0 = \pm 1$.

Slide $x_0$ between $-3$ and $3$ and watch the red tangent rotate against the blue cubic $y = x^3 - 3x$. The lab reports $f(x_0)$, $f'(x_0)$ and marks the critical points $x = \pm 1$ where slope is zero.

Slope tells you the instantaneous rate of change. In economics, $f'(x)$ is marginal cost; in physics, it's velocity. JEE Main and Advanced questions about tangents, normals, monotonicity and extrema all start from $f'(x_0)$. This lab is the visual core of differential calculus.

Tangent has slope $f'(x_0)$. The normal is perpendicular to the tangent, so its slope is $-1/f'(x_0)$ (when $f'(x_0) \ne 0$). Students forget the negative reciprocal. The lab shows only the tangent — sketch the normal mentally as a check.