What is a derivative of a function?How is it related to limits?

[Millan]

I know application of derivative( if we draw tangent at a pt then it gives slope of point).
I want to know exact definition.Also first principle uses limits.How is limit and derivative connected?
Also , is there a function that is continuous but not differentiable?

[TI]

derivative is the gradient of a function, it is used to obtain/analyse turning points and how they behave in relation to the function ... many realistic situations can be modelled as functions and its derivatives, especially those involving rates of change

limits are when you hypothetically/intentionally assign a specific value for a domain (i.e the x) and see how the function behaves as it progresses (not to be confused with integral limits)

[Hawke]

dy/dx := lim(h->0) [f(x+h) - f(x)]/h

A derivative is defined by the limit of the slope formula of a secant line as the secant line collapses into a tangent line (imagine sliding the second point of the secant line towards the first point, it becomes a tangent line to the first point).

[Hawke]

The derivative is a limit because we are trying to narrow down dx as much as possible. The derivative is essentially a slope, correct? The slope of any line is dy/dx, correct? Well in order to find the EXACT slope of a curve at any given point, the change in x, dx, needs to be as small as possible. Therefore, we set it to 0. Unfortunately dy/0 cannot be evaluated as it does not exist. Therefore we take the limit, to see how the y changes, dy, as the change in x, dx, gets infinitesimally small.

That is a simple explanation and, as I'm sure you know, the exact definition involves the limit as h approaches 0 of [(f(x+h) - f(x)]/h. This cannot be evaluated because the denominator is 0, so we must take limits.

To answer your question, yes. A cusp. An example of a cusp would be the absolute value graph. It is continuous, though not differentiable.

[Hawke]

Hello Zohair, yes the function y = |x| is continuous but not differentiable. The graph drawn would look like /\. There is continuous drawing. But at the peak there is a sharp point. If we have to draw tangents then innumerable tangents could be drawn there. Hence not differentiable.
If you have an equation y = 4x + 5. This stands for a straight line. So its derivative dy/dx = 4 which is the slope of the line. You know that, good.
If we have x^2 = y then dy/dx = 2x
Here slope m depends on x, so as x varies m also varies. So there can't be a straight line. It has to be a curve. And so the curve is that of a parabola. Facing upward with y axis as the axis of the parabola.
First realize the meaning of dx. dx is the derivative of x. It is obtained as delta x tends to zero. Note down. "tends to". Delta x is not equal to zero but nearing zero. Then it is denoted as dx.
delta y / delta x becomes dy/dx as delta x tending to zero.
This is the meaning behind the derivative concept.
Now let us see about the limits. Suppose a line y = x+4 is drawn. It would be passing through (-4,0) and (0,4). This makes an angle 45 deg with the positive x axis.
Now I would like to know about the area bound by this line (topside), x = 3 (left side) x=5(right side) and y=0 ie x-axis (down side).
Then I use the integral technique. For delta x gap along x axis at x distance, there will be y height parallel to y axis. Hence the area bound by y, dx will be ydx.
Such strips could be imagined drawn with the region in millions and millions. Hence the total area is got by integrating this y dx. within the limits x = 3 and x = 5.
So y dx or xdx + 4dx. Integrating we get x^2 / 2 + 4 x
Now applying limits, 25/2 + 20 - (9/2 + 12) = 8 + 8 = 16 sq units.
Hence we need limits. It may be even right from x = -2 to x = 9
In this case we would have enclosed areas one below x axis (from -2 to 0) and the other above x axis (from x=0 to 9). Hence we have - integral and + integral.
Another simple explanation. If suppose a table is at a height 80 cm right from the floor. A box of height 40 cm is kept on the table. Now the potential energy for height on the table will be mg(0.80). And that on the box will be mg (1.20). Now the potential difference is upper - lower ie mg (1.20 - 0.80)
The same limiting is undertaken in case of integration.