# 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 x^{b}, its derivative is simply
b ´ x^{b–1}.

The derivative
of _{} is _{}.

2. For any ax^{b} its derivative is b ´ ax^{b–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 3x^{4} is 12x^{3}.

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 2x^{4}
+ 3x^{2} – 3x + 2 is 8x^{3} + 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 e^{x} is e^{x}.

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.