Question

Derivatives Tutorial

Answer 1

For example we have a graph of function f(x). We see that on some intervals it increases, on some decreases and has maximum and minimum points. To show the intervals where function decreases and increases, there is a special function, which is called derivative of function. Derivative of function f(x) shows the velocity of change of velocity of function. When f(x) decreases its velocity is negative. When it increases - the velocity is positive. The min and max points and intervals where f(x) is horizontal, the velocity of f(x) is 0 - the value of function doesn't change. Mathematically derivative is defined as: f'(x)=lim_(c->0) (f(x+c)-f(x))/c
If we substitute a value instead of a, we will get a slope of function f(x) at point x = a. If we don't substitute any value for a, we will get an equation of derivative of function f(x).
It is often useful to use the table of derivatives of simple functions when solving derivatives of big functions.
We learned how to find derivatives of functions if we know the equation of function and we need to find its derivative. However, not all problems require finding the equation of derivatives. For example we are given a graph, and we need to graph its derivative. To do this follow these steps:
1) Identify the min and max points. Since they are min and max points the derivative at these points will be 0.
2) Since the function is continuous and doesn't have sharp turns its derivative will also be continuous. Now identify the increasing and decreasing intervals of function.
3) Before first max point the function only increases, but since it eventually turns down, its speed is slowing down. So draw a smoothly decreasing line, which has its values only y>0.
4) Between max and min point function decreases, so its speed is negative. At first it accelerates, but then slows down, so draw a line which has values y<0 and which has a negative slope at first and then has a positive slope.
5) After the min point, the graph accelerates and increases, so draw a line with increasing slope and with values y>0.
6) The derivative of this function should have form of parabola, opening up.

Some properties that show where the function doesn't have derivative.
1) In points with sharp turn, function f(x) is not differentiable.
2) In points when function is not continuous, function f(x) is not differentiable.

When solving different problems on physics we usually use words "area", "volume", "distance", "velocity" and "acceleration". Although these terms measure different things, they have something in common between each other. The area is derivative of volume, distance is derivative of area, velocity is derivative of distance and acceleration is derivative of velocity. Actually, what is derivative of function?
Derivative of function is another function which shows the speed of increasing and decreasing of given function. If we have a graph of changing of distance from time, the derivative will show the speed of changing of distance from time, or as it is said in physics, will show velocity of object. If we graph the second derivative from graph showing the relationship between distance and time, we will get a graph of changing of acceleration over time. The acceleration is speed of changing of velocity of object. When we take derivative from graph showing relationship of velocity from time, we get the graph of velocity of changing speed of object, which is called acceleration.

Answer 2

\(\Large f'(x)=\lim_{c \rightarrow 0}\frac{ f(x+c) -f(x)}{ c }\) this is the definition of derivative

Answer 3

@ksanka awesome hob!! :D I have no idea of this stuff but I'm pretty sure that it's gonna be very helpful :) im glad u took ur time to make this u r awesome !! :)

Answer 4

lol

my sister is making awesome tutorail as well as i am.

Answer 5

Welp u guys are awesome!! :)

Answer 6

Job*** ^^

Answer 7

Hey, make sure you submit it to Abhisar or go to openstudytutorials@weebly.com and submit it for consideration for Top Ten Tutorials!

Answer 8

make sure you cite it properly too.

Answer 9

Great job on derivatives.

Answer 10

thnx

Answer 11

great explanation of the beginning of derivatives