Partial Derivative
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This example illustrates the concept of partial derivatives, which are the rates of change of a multivariable function in a specific direction. By constructing the tangent lines of a curve, the geometric meaning of partial derivatives can be observed. Commands used include IntersectPath, Derivative, and CurveCartesian. Suitable for teaching multivariable calculus in university mathematics courses.
GeoGebra Commands & Parameters
IntersectPath(p, a) → eq1Derivative(a(x, yc)) → dfxDerivative(a(xc, x)) → dfyCurveCartesian(t, yc, a(xc, yc) + (dfx(xc) * (t - xc)), t, -10, 10) → b
AI Drawing Prompt Reference
Draw a 3D surface representing a multivariable function, and illustrate the geometric meaning of partial derivatives. First, construct the intersection curve of the surface with a vertical plane parallel to the x-axis, yielding a space curve. Then compute the partial derivatives of the function with respect to x and y. At a chosen point on the surface, draw a tangent line in the x-direction, lying in the plane where y is fixed, and tangent to the intersection curve. Finally, display the surface, the intersection curve, and the tangent line together to show how the partial derivative measures the instantaneous rate of change in that direction.
The prompt above was compiled from the original drawing commands and is for reference only.