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I've seen several answers but non of them do not answer what I am looking for, therefore I am asking here again,

According to my understanding,

Slope, $\frac{\partial f}{\partial x}$, is generally applicable when only 2 variables are in consideration. The slope is the tangent or the derivative to the function's curve that connects the 2 variables, i.e., the measure of the rate of change of a function f(x) with respect to the x.

Gradient is the transpose derivatives or just the vector of partial derivatives.

Consider $f(x,y) = y-x$

Gradient of this $\nabla f(x,y) = \frac{\partial f(x,y)}{\partial x}i + \frac{\partial f(x,y)}{\partial y}j = -i+j$

Question:

Since the slope or tangent line and gradient are perpendicular to each other is there a way to prove this?

Ted Shifrin
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GPrathap
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  • Let $f:\mathbb{R}^n \to \mathbb{R}$ differentiable and $\mathbf{v}\in \mathbb{R}^n$ fixed, then for any unitary vector $\mathbf{u}$ the quantity $\nabla f(\mathbf{v})\cdot \mathbf{u}$ defines de slope of the graph of $f$ at $\mathbf{v}$ in the direction $\mathbf{u}$. Indeed $\nabla f(\mathbf{v})\cdot \mathbf{u}$ is just the directional derivative of $f$ at $\mathbf{v}$ in the direction $\mathbf{u}$ – Masacroso Feb 22 '23 at 06:25
  • @ParclyTaxel yeap, now I understand whats going on there, I did not know about level curves and level sets. Thank youuu – GPrathap Feb 22 '23 at 06:58
  • People various places in the world (e.g., much of Europe?) use the word gradient as a synonym of slope. "Slope or tangent line"? No. Slope is a number, not a line. The gradient is perpendicular to the tangent line of a level curve, yes. – Ted Shifrin Feb 23 '23 at 01:38

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