Derivatives
Derivatives are a major part of any high school mathematics course. The derivative is really just the slope of the curve of a function, although it has a wide range of uses. Differential calculus, of which derivatives is a part, gives us an exact way of calculating the slope of a function at a point, without having to draw tangent lines and work out slopes graphically.
There are a number of rules for finding the derivative of a function, and you really have to learn them. They are as follows:
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1. For any xb, its derivative is simply b ´ xb–1.
The derivative
of 
is 
.
2. For any axb its derivative is b ´ axb–1. The power is put out in front of the expression, and then the power is reduced by 1. Remember anything to the power 1 is just itself, and anything to the power 0 is 1.
The derivative of 3x4 is 12x3.
The derivative of 3x–4 is –12x–5
3. When you have terms separated by ‘+’s or ‘–’s you can find the derivative of each term then add them together to find the derivative of the whole function.
The derivative of 2x4 + 3x2 – 3x + 2 is 8x3 + 6x – 3
Note that the 2 just disappears. All constants (numbers by themselves without algebraic symbols) disappear when you find the derivative.
4. The derivative of sin x is cos x
5. The derivative of cos x is –sin x (note the change in sign).
6. The derivative of ex is ex.
7. The derivative of ln x is 1/x, for x > 0. There is no ln x when x £ 0 (ln is the natural logarithm)
Note that the derivative can be written a few different ways:
·
If you have 
, then the derivative can be written
·
If you have 
, then the derivative can be written
· If you have a function with ‘x’ as the variable being differentiated, the derivative can be written as:
· ‘Something’ is the function which has ‘x’ in it.
There are also three rules that are handy for finding derivatives when the functions are more complicated.
