Geometrical meaning of derivatives

[nil1996]

I don't understand what the derivatives really mean?I know that they are the slope of the tangents drawn to a function.But see for example we have a function f(x)=x²
The derivative of this gives us '2x'. But what does '2x' mean?If i draw a graph of f(x)=2x what does it give me?what should i understand from that graph?

please help

 

[Millennial]

Plot the curve $y=f(x)$ on the Cartesian plane. Now, pick a point $(x_0,f(x_0))$ from the curve you plotted and draw the tangent to the curve at that point. The slope of the tangent line will be $f'(x_0)$. As a solid example, take the function you provided, $f(x)=x^2$. If you wanted to know the slope of the tangent line at $x=3$ for this curve, i.e the point $(3,9)$, you would plug in $x=3$ to the derivative and get $2\cdot 3 = 6$. This value is also the instantaneous rate of change of the function.

 

[Stephen Tashi]

The derivative of this gives us '2x'. But what does '2x' mean?If i draw a graph of f(x)=2x what does it give me?what should i understand from that graph?

I don't think you shoud expect to see any simple geometric relationship between the graphs of fix) and its derivative. There are qualitative relationships, such as "when the derivative is positive at a point x, the function f(x) is increasing at that point", "when the derivative is zero at x, the function f(x) is instanteously flat, such as at a peak or valley".

You can do what Millenial suggests and interpret the graph of the derivative to get the slope of the tangent line at x and then go to the graph of f(x) and verify that this number is indeed the slope. But don't expect to see any straightforward geometric insight like "if I drawn a tangent to the function at x and extend the line till it hits the graph of the derivative and erect a perpendicular there and ...".

It's typical in math to use a "pointwise" operation to define a function by considering the point to be a variable. The derivative is defined as an operation that produces a number when carried out on the function f(x) at a given point x. But since we can vary x we can say this operation defines a function of x.

For example, for the function f(x) = x^2 we could define the pointwise operation at the point x by "take the average of f(x-1) and f(x+1)". Since this operation is defined at each point x, you could graph g(x) as a function. Graphing f(x) and g(x) on the same plot wouldn't necessarily reveal any pleasing geometric interpretation - although I haven't tried it.

 

[WannabeNewton]

The derivative allows you to find the "best linear approximation" of the function near a given point. In single variable calculus, if you graph a function and consider a specific point you will notice that the tangent line passing through that point (with slope given by the derivative of the function at that point) will approximate the graph for points sufficiently close to the original point. This is something you will see come up over and over, beyond single-variable calculus (except in higher dimensions and more abstract terms).