Implicit Differentiation is an approach to taking derivatives that uses the chain rule to avoid solving explicitly for one of the variables.
For example, if \( y + 3x + 8 = 0 \), we could solve for \( y \) and then differentiate:
However, we could also simply take the derivative of each term with respect to in place:
The second approach is known as implicit differentiation.
Given what is at the point ?
Since , differentiating, we have:
Thus at the point is .
If what is ?
Taking the derivative of every term with respect to gives us:
Now, we can isolate all of the on the left: