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I need your help, my question is what is the difference between tangent and slope of tangent ?

A clear example would be appreciated.

Thank you.

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The tangent to a curve at a point is a straight line just touching the curve at that point; the slope of the tangent is the gradient of that straight line.

Here's a picture to help.

enter image description here

The green line is the tangent line to the point $(1,1)$. It is a geometric object.

The slope at $(1,1)=2$ (if you want to know why, it's to do with differentiating, or finding the gradient of the tangent at a point). This is a number.

So, to conclude: the tangent's a line; the slope is a number.

Qualitatively, your question is equivalent to the following:

"What's the difference between a hill and the steepness of a hill?".

To answer your second question (in the comments): $$\tan(\theta)=\frac{\textrm{change in y}}{\textrm{change in x}}=\frac{\Delta y}{\Delta x}=\rm{gradient}$$ (we can see this from the right-angled triangle, if we use some basic trigonometry).

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  • $\begingroup$ does that mean that when we compute the tan(theta) we get the slop of the tangent. $\endgroup$ – Mohamad Jun 14 '14 at 21:49
  • $\begingroup$ What is $\theta$? $\endgroup$ – beep-boop Jun 14 '14 at 21:50
  • $\begingroup$ theta is the angle $\endgroup$ – Mohamad Jun 14 '14 at 21:53
  • $\begingroup$ If $\theta$ is the angle between the tangent and the horizontal, then, yes: $\tan(\theta)=\rm{slope}$. $\endgroup$ – beep-boop Jun 14 '14 at 21:53
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    $\begingroup$ Picture made with $\LaTeX$ of 1950. $\endgroup$ – Relure Jun 14 '14 at 21:55
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The English word 'Tangent' also means the tangent line (at a point of a circle, say), and its slope is the numeric value $\tan(\alpha)$ where $\alpha$ is the angle of the tangent line and the right wing and (any parallel to) the $x$-axis.

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