[Math] what is the difference between tangent and slope of tangent

Question

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.

Answer 1

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.

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).