Math 22 Implicit Differentiation

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Implicit Differentiation

Consider the equation . To find , we can rewrite the equation as , then differentiate as usual. ie: , so . This is called explicit differentiation.

However, sometimes, it is difficult to express as a function of explicitly. For example:

Therefore, we can use the procedure called implicit differentiation

Guidelines for Implicit Differentiation

 Consider an equation involving  and  in which  is a differentiable function of . You can use the steps below to find .
 1. Differentiate both sides of the equation with respect to .
 2. Collect all terms involving  on the left side of the equation and move all other terms to the right side of the equation
 3. Factor  out of the left side of the equation.
 4. Solve for  by dividing both sides of the equation by the left-hand factor that does not contain .

Example: Find of





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