# Graph od Derivative

1. Jan 12, 2005

Hello all

Given the graph of the function $$y = x^4 + 2x^3 + x^2 + 6$$ graph its derivative.

In other words, how would you graph the derivative soley based on the function's graph? Would you just draw a horizontal line if the slope is 0, etc..?

Thanks

2. Jan 12, 2005

### dextercioby

This is a weird question.From the graph of the function,of that function actually,u could tell where the derivative is zero and the signs it has on various intervals between its zeros.And nothing more...

Daniel.

3. Jan 12, 2005

### BobG

No. The points where the slope of the original function is zero are the points where the graph of your derivative crosses the x-axis. If the slope of the function is positive, your derivative is above the x-axis. If the slope of the function is negative, your derivative is below the line. If you can pick out the inflection points (where the curve changes shape), these are the points where your derivative hits a local maximum or minimum.

The function they give you can be a little tricky if you use a graphing calculator and don't look at it closely. You actually have two local minimums and a local maximum - these are the three points where your derivative crosses zero. Actually determining your critical points is always your best for graphing the derivative of the function.

4. Jan 12, 2005

### danne89

Have I totaly missunderstud this. Can you really graph a derivate, or just a tangent or are they synonyms of each other in this context?

5. Jan 12, 2005

### dextercioby

The velues of the derivative (seen as a function:"derivative (x)") are the values the slope of the original graph can take.
U can't really graph a derivative knowing the graph of the original function.

Daniel.

6. Jan 12, 2005

### danne89

So, when you refere to the derivate you refere to the function and when you refere to the tangent you specificly means the line?

7. Jan 12, 2005

### dextercioby

Yes,the derivative is the function and its "y" values (the values from its range/image) are the values for the slope of the function.And they correspond to the same domain:e.g."f" has the slope at the point (7,f(7)) f'(7) .

Daniel.

8. Jan 12, 2005

### vincentchan

yes... and the slope of tangent is the value of derivative(of the function)...

9. Jan 13, 2005

### BobG

Yes. A simple way to illustrate the difference.

Your velocity is the derivative of your position. Your instantaneous velocity gives the rate of change in your position at a given instant. Your velocity at that exact instant would be the equivalent of the slope of your tangent line.

But, how about if you drive at 60 kph for 30 minutes, stop at a stop light for 2 minutes, then turn onto the highway and drive 145 kph for an hour, get pulled over by the police and remain motionless for the 30 minutes it takes the policeman to write your ticket, then despondently drive 90 kph for the 45 minutes it takes to complete your trip.

The graph of your derivative is a lot more than just one tangent line. It includes all the different velocities you travelled, including all the velocities you passed through when accelerating from 60 to 0 to 145, etc.