# 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:

## Differentiation Rules

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.