Derivatives

=**Four ways to derive!**=

4. Chain rule
=**Other important factors when deriving!**=

__Power Rule:__
1. Multiply exponent in front 2. Subtract "1" from exponent

Example:

[[image:CodeCogsEqn_(8).gif]]
See that the exponent changes the coefficient to 15x3. Then the final answer has a 2 exponent because you took that 3 and subtracted 1.

You can also use the power rule when adding and subtracting.

Example: Find the derivative of:

1. Derivative of he first piece 2. Times original second piece 3. Plus original first piece 4. Time derivative of the second piece
 * __Product Rule:__ **

Find the derivative of:


Sometimes you will need to use both the power and the product rule in order to figure out the derivative.

Example: Find the derivative of:

Notice that in order to finish the derivative you had to first use the power rule to solve for 4x^3.

**__ Quotient Rule: __**
Bottom: original denominator squared Top: 1. original denominator 2. times derivative of numerator 3. minus derivative of denominator 4. times original numerator

Find the derivative of:


When using the quotient rule, all of the derivative rules can apply to this, the power, the product and the chain rule.

__ Chain Rule: __
1.Derivative of the first piece 2.Keep original inside piece 3.Multiply by the derivative of inside piece

[[image:CodeCogsEqn.gif2.gif]]
When using the chain rule it can be tied in with all the other rules, just like the ones I previously stated. Also, the chain ruled is used with the implicit differentiation rule.

=__** Trig Rules: **__= Trig rules are rules that need to be memorized in order to make derivatives much easier. Trig has to do with sin, cos, tan, etc. This rules will help you when solving those tricky derivatives.

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==** Remember that all the trig rules listed can be tied into all of the derivative rules. I even listed examples where you needed to use them. A helpful way to remember the sin and cos rules is to remember this simple list: **== =** sin **= =** cos **= =** -sin **= =** -cos **= == **This list works where to find the derivative, you move downwards, for example the derivative of sin, would be one down, cos. This list is also helpful when solving for the anti-derivative, all you need to do is go up the list. For example, the anti-derivative of -sin is cos.** ==

ALL TOGETHER!!
All of these rules and trig rules can all be tied to together like these ones:

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