How tangent line gives us a slope at one point?

Question

As we know that slope is the rate of change of one quantity with respect to the other. So we do always need two points to find the slope. But then its confusing when they say that tangent line gives us a slope at one point. If its just one point and tangent line is just touching only that one point then how the tangent line is giving us its slope. Does that mean that tangent line gives us a slope of a point and its neighbouring point which is so much close to it so that they give a sense of a straight line between them and the tangent line is giving us the slope of these two points? Please explain the concept only.

Answer 1

The slope of the tangent line is the limit of the slope of the two points as they get arbitrarily close together. Hence the definition

$$f'(c)=\lim_{v\to c}\frac{f(v)-f(c)}{v-c}.$$

Note that for $f'(c)$ to be meaningful, the above limit must exist, which is why a function is not differentiable if it has a discontinuity or a sharp cusp.

Answer 2

Consider a a real valued function $\;y=f(x) \;\;\; $ defined on some open interval of the real line. Suppose $\;c\;\;\;$ is a given point of the domain. If the derivative $\;f^{'}(c) \;\;\;$exists at the point c, then we can correspond this number to the line in the plane with slope$\;f^{'}(c) \;\;\;$ and passing through the point$\;P=(c, f(c)) \;\;\;$ This can be shown as the tangent line of curve at P. Thus a unique tangent line exists at the point P of the curve representing the graph of the function. Conversely, if we can have a unique tangent line at the point P=$(c, f(c)) \;\;\; $then we can show, by the basic definition of derivative of the function f at the point c that the slope of the tangent line is precisely $\;f{'}(c) \;. \;\;$ This follows from the fact that the slope of the secant line joining two points Q, R of the graph enclosing the point $\;(c, f(c)) \;\;\;$ tends to the slope of the tangent at the point $\;(p, f(c)) \;\;\;$ as the length of QR tends to zero. This interpretation holds true only for the derivative at a single point of the domain.