The derivative of a single-variable function, ( f(x) ), measures how ( f ) changes with respect to small changes in ( x ). For example, if ( f(x) = x^2 ), the derivative is: [ f'(x) = \frac{df}{dx} = 2x ] This derivative tells us the rate of change of ( f ) as ( x ) changes. In single-variable functions, we only have one input variable, so we only need one derivative to describe the function's slope at any given point.
- The derivative applies to single-variable functions.
- It tells us how the function value changes as the single variable changes.
- It results in a single value representing the slope at a particular point.
When dealing with functions of more than one variable, such as ( f(x, y) ), we use partial derivatives. A partial derivative represents how the function changes with respect to one variable while keeping the other variables constant.
For a function ( f(x, y) ), we can find:
- The partial derivative with respect to ( x ): ( \frac{\partial f}{\partial x} )
- The partial derivative with respect to ( y ): ( \frac{\partial f}{\partial y} )
These partial derivatives tell us the rate of change in the ( x ) and ( y ) directions, independently. This is useful in functions with multiple variables, where we might want to know the effect of changing one variable while holding others constant.
Consider the function: [ f(x, y) = x^2 + y^2 ] The partial derivative of ( f ) with respect to ( x ) (treating ( y ) as constant) is: [ \frac{\partial f}{\partial x} = 2x ] The partial derivative of ( f ) with respect to ( y ) (treating ( x ) as constant) is: [ \frac{\partial f}{\partial y} = 2y ] These partial derivatives tell us:
- How ( f ) changes as ( x ) changes, with ( y ) held constant (( 2x ) gives the rate of change in the ( x )-direction).
- How ( f ) changes as ( y ) changes, with ( x ) held constant (( 2y ) gives the rate of change in the ( y )-direction).
The gradient is a vector that combines all partial derivatives of a multi-variable function. For a function ( f(x, y) ), the gradient ( \nabla f ) is: [ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) ] This vector points in the direction of the steepest ascent of ( f ) and gives the rate of increase in that direction.
For our function ( f(x, y) = x^2 + y^2 ), the gradient is: [ \nabla f = (2x, 2y) ] So, at any point ( (x, y) ), the gradient points in the direction of maximum increase of ( f(x, y) ), and its magnitude ( | \nabla f | ) gives the rate of that increase.
- The gradient is a vector formed by the partial derivatives of a multi-variable function.
- It combines the information from each partial derivative, giving both the direction of steepest ascent and the rate of increase in that direction.
- In optimization, following the negative gradient direction leads to the steepest descent, making it useful for finding minima of functions.
| Concept | Single-Variable Function | Multi-Variable Function |
|---|---|---|
| Derivative | Measures rate of change of ( f(x) ) as ( x ) changes. Single value representing slope. | Not directly applicable, as multi-variable functions require partial derivatives. |
| Partial Derivative | Not applicable in single-variable functions, since only one variable exists. | Measures rate of change of ( f(x, y, \dots) ) with respect to one variable, keeping others constant. |
| Gradient | In single-variable, gradient is the derivative itself. Points in the direction of maximum increase of ( f(x) ) (positive or negative slope). | Vector of all partial derivatives. Points in the direction of the steepest ascent of ( f(x, y, \dots) ). Useful in optimization. |
In essence:
- Derivatives describe changes in single-variable functions.
- Partial derivatives describe changes in each direction in multi-variable functions.
- Gradients combine all partial derivatives into a vector, pointing in the direction of steepest ascent in multi-variable functions.