Insights What is a Tangent Line? A 5 Minute Introduction

[Greg Bernhardt]

Definition/Summary
The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.
More generally, the (n-1)-dimensional tangent hyperplane to an (n-1)-dimensional surface in n-dimensional space at a particular point has the same Gradient as the surface has at that point.
So if A\,=\,(a_1,a_2,\cdots a_n) is a point on a surface defined by the equation F(x_1,x_2,\cdots x_n) = 0, then the tangent hyperplane to the curve through A is \frac{\partial F}{\partial x_1}\arrowvert_A(x_1 – a_1)\,+\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_2 – a_2)\,+\,\cdots\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_n – a_n)\,=\,0
If a curve in n dimensions is defined using a parameter t as A(t)\,=\,(a_1(t),a_2(t),\cdots a_n(t)) , then its tangent is:
(x_1 – a_1) / \frac{da_1}{dt}\,=\,(x_2 – a_2) / \frac{da_2}{dt}\,=\,\cdots\,=\,(x_n – a_n) / \frac{da_n}{dt}
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[fresh_42]

See also https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ and the list at the beginning in https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ where I have listed a couple of diffferent views of how a derivative can be seen. Slope wasn't even among it!