Calculus
DIFFERENTIATION
For differentiation, we will give you a table with functions, and their corresponding derivatives. Then we will show you a general formula for taking the derivative of a function.
| The Function | A Derivative |
| f(x) = x | f ' (x) = 1 |
| f(x) = 4x | f ' (x) = 4 |
| f(t) = 4t | f ' (t) = 4 |
| f(s) = 4s | f ' (s) = 4 |
| f(x) = x2 | f ' (x) = 2x |
| f(s) = s2 | f ' (s) = 2s |
| f(x) = x5 | f ' (x) = 5x4 |
| f(x) =2 x5 | f ' (x) = 10x4 |
| f(x) =3 x5 | f ' (x) = 15x4 |
After looking at the above examples, you may have noticed some trends:
- In taking the derivative, the superscript always decreases by one.
- The coefficient in front of the variable is multiplied by the number of the superscript. Recall that if no number is listed, then the coefficient is 1.
If f(x) = kxn
then
f ' (x) = (kn)xn-1
We've used the notation f ' (x) for derivatives because it is easy to type into the computer. However, this notation is not the best. What if we have the function
which contains two variables?
We could take the derivative with respect to x:
df(x,y)
-------- = 2xy3
dx
We could take the derivative with respect to y:
df(x,y)
-------- = 3x2y2
dy
INTEGRATION
For integration, we will give you a table with functions, and the functions that result after integration. Then we will show you a general formula for integrating a function.
| The Function | Result of Integration |
| x |
1 2
- x
2
|
| 2x | x2 |
| 4x | 2x2 |
| x3 |
1 4
- x
4
|
| 5x3 |
5 4
- x
4
|
| 7x2 |
7 3
- x
3
|
For more in depth information, try the
MIT Calculus Index
Ripley's Escape from the Nostromo- A calculus problem
Return to Master Alchemist Index
Last Revised 02/04/98.
Copyright ©1998 by William L. Dechent. All rights reserved.