Partial Derivative Visual: Tangent Planes and Directional Slopes on a Surface

①This example demonstrates the geometric meaning of partial derivatives by showing a 3D surface with a movable point, where the slope along the x- and y-directions is visualized as tangent lines or a tangent plane.②Core mathematical concepts include partial derivatives, directional derivatives, tangent planes, limits, and multivariable calculus; the construction also references integers, natural numbers, conditional probability, and cosine, but the main focus is on differentiation.③Construction highlights: from the summary, objects a, f, b, A, g, h are used—likely f is a bivariate function surface, A is a point on the surface, g and h are curves or planes representing the partial derivative slices, and sliders or input fields control the point coordinates to update the displayed slopes.④This is suitable for university-level calculus or multivariable calculus courses, as well as advanced high school students or teacher demonstrations, helping learners connect abstract derivative formulas with visual 3D representations.