Difference between revisions of "Derivative"
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The '''derivative''' of a [[function]] is defined as the instantaneous [[rate]] of change of the function with respect to one of the [[variable]]s. Note that not every function has a derivative. | The '''derivative''' of a [[function]] is defined as the instantaneous [[rate]] of change of the function with respect to one of the [[variable]]s. Note that not every function has a derivative. | ||
| + | == Notation == | ||
| + | The derivative of f(x) can be expressed in several ways including: | ||
| − | == | + | * <math>\frac{d}{dx}</math> |
| + | * <math>f'(x)</math> | ||
| + | * <math>f'</math> | ||
| + | |||
| + | == Finding the Derivative == | ||
For any constant, the derivative is 0. | For any constant, the derivative is 0. | ||
| − | For any monomial <math>nx</math>, the | + | For any monomial <math>nx</math>, the derivative is n. |
Note that when we take the derivative of any polynomial <math>a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0</math>, we can take the derivative of each addend and then add these to find the derivative of the polynomial. | Note that when we take the derivative of any polynomial <math>a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0</math>, we can take the derivative of each addend and then add these to find the derivative of the polynomial. | ||
| − | The chain rule states that the derivative of any <math>ax^n</math> is <math>anx^{n-1}</math> | + | The [[chain rule]] states that the derivative of any <math>ax^n</math> is <math>anx^{n-1}</math> |
| + | |||
| + | To find the derivative of <math>f(x) \cdot g(x)</math> we cannot do what we did with addition. We must instead use the [[product rule]]: <math>(f(x) \cdot g(x))' = f'g + g'f</math> | ||
| + | |||
| + | The [[quotient rule]] states that <math>(\frac{f}{g})' = \frac{f'g - fg'}{g^2}</math> | ||
The following pages provide additional information on derivatives. | The following pages provide additional information on derivatives. | ||
Revision as of 22:04, 9 September 2006
The derivative of a function is defined as the instantaneous rate of change of the function with respect to one of the variables. Note that not every function has a derivative.
Notation
The derivative of f(x) can be expressed in several ways including:
Finding the Derivative
For any constant, the derivative is 0.
For any monomial 
, the derivative is n.
Note that when we take the derivative of any polynomial 
, we can take the derivative of each addend and then add these to find the derivative of the polynomial.
The chain rule states that the derivative of any 
is 
To find the derivative of 
we cannot do what we did with addition. We must instead use the product rule: 
The quotient rule states that 
The following pages provide additional information on derivatives.
- Notations
- Formal definition of the derivative
- Formulas
See also
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