Implicit differentiation

Summary

• Given an implicit relationship between two variables \(x\) and \(y\) , you can find the derivatives \(dy/dx\) (also called \(y'\) ) using the method of implicit differentiation .
• Differentiate both sides of the equation defining the relationship.
• Differentiate all \(x\) -variables as “ usual ” . \(x^2\) to \(2x\) , \(log x\) to \(1/x\) and so on.
• Differentiate all \(y\) -variables by treating them as functions of \(x\) .
• For example, \(y^2\) must be viewed as a composite function with derivative \(2yy'\) .
• The differentiated equation will be a new equation. Solve this for \(y'\) . This is \(dy/dx\)
• The same method can be used to find the second derivative, \(d^2y/dy^2\) or \(y^{''}\) . Differentiate the equation once more and solve for \(y^{''}\) .
• The same method can be used to find the derivative \(x'\) or \(dx/dy\) . Differentiate both sides but switch the roles for \(x\) and \(y\) and solve for \(x'\) .