# what is the difference between tangent and slope of tangent?

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.

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).

• does that mean that when we compute the tan(theta) we get the slop of the tangent. Jun 14, 2014 at 21:49
• What is $\theta$? Jun 14, 2014 at 21:50
• theta is the angle Jun 14, 2014 at 21:53
• If $\theta$ is the angle between the tangent and the horizontal, then, yes: $\tan(\theta)=\rm{slope}$. Jun 14, 2014 at 21:53
• Picture made with $\LaTeX$ of 1950. Jun 14, 2014 at 21:55

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.