Derivative of Implicit functions
- Implicit functions are equations where the relationship between variables is not explicitly stated. Instead, it's presented as an equation involving both variables.
- Typically represented as or , where or could be any expression involving both and .
Derivative of Implicit Functions:
- Finding the derivative of an implicitly defined function involves using implicit differentiation, treating as a function of even when isn't explicitly written as a function of .
Steps to Find Implicitly:
- Differentiate both sides of the equation with respect to :
- Apply the chain rule whenever appears.
- Solve for :
- Collect terms involving on one side of the equation.
Given the equation :
Differentiate both sides with respect to :
Solve for :
- Pay attention to chain rule usage:
- Every term involving must be multiplied by when differentiating with respect to .
- Isolate the derivative:
- After implicit differentiation, isolate if possible to express it explicitly.
- Watch for implicit relationships:
- Implicit derivatives might reveal relationships between and not easily seen in explicit functions.
- After finding the first derivative , further derivatives can be found by differentiating the equation again and solving for or higher derivatives if needed.
- Sometimes it's helpful to express an implicitly defined function in parametric form, where and are defined separately as functions of a parameter.