The Connection Between Slope and Derivatives: Understanding the Relationship

[Swetasuria]

If there is an equation for a curve, its derivative will be the slope of the tangent.
Also, the derivative of a function is the limit of its slope.

What I understand from this is that (slope of tangent)=(limit of the same slope)

But this is wrong (right?). Please explain the mistake here.

 

[Dick]

The derivative is the limit of the difference quotient. You can call this limit the slope. I have no idea what you mean by 'limit of the slope'.

 

[Swetasuria]

The derivative is the limit of the difference quotient. You can call this limit the slope. I have no idea what you mean by 'limit of the slope'.

Even so, I still can't understand the mistake I made.

 

[Muphrid]

You're not taking the limit of the slope of tangent lines. You're taking the limit of the slope of secant lines. The secant line between points A and B has a slope that, in the limit that A and B come together, is the tangent line slope.

 

[Dick]

Even so, I still can't understand the mistake I made.

What mistake? The limit of the derivative is not necessarily the derivative of the limit, which is the best way I can think of to make sense of your question. Take x^2*sin(1/x^2). It has a derivative at x=0. The limit of the derivative as x->0 doesn't exist.

 

[HallsofIvy]

Your mistake is talking about the "slope" of a function at all. "Slope" is only defined for lines. If a function is linear, then its graph is a straight line and so its graph (not the function) has a slope. If a function is not linear, then its graph is NOT a straight line and neither the graph nor the function has a "slope". We can, at each point, draw a line tangent to the graph and talk about its slope.