Differentiability and the Total Derivative Differentiability measures how a function's rate of change changes with respect to changes in its input variab...
Differentiability measures how a function's rate of change changes with respect to changes in its input variables.
Total derivative, on the other hand, represents the instantaneous rate of change of a function.
In simpler terms:
Differentiability: tells us how a function's slope changes at a given point.
Total derivative: tells us how the function's value changes as its input changes.
Examples:
Differentiability:
The derivative of the function f(x) = x^2 is 2x.
This means that the function's slope is 2 for any value of x.
Total derivative:
The total derivative of the function f(x) = x^2 is 4x.
This means that the function's value increases by 4 units for every unit increase in x.
Relationship between the two:
The total derivative is the derivative of the function in the direction of the change in the input variable.
In other words, it tells us how the function changes as we move along the input variable.
Applications:
Differentiability is used in various economic models to analyze and predict how a function's output changes in response to changes in its inputs.
For example, the marginal product of a firm is the derivative of its total profit function with respect to its input.
Key takeaways:
Differentiability tells us how a function's slope changes, while the total derivative tells us how the function's value changes at a given point.
The total derivative is the derivative of the function in the direction of the change in the input variable.
These concepts are essential for understanding the behavior of functions and solving economic problems involving optimization and marginal analysis