In implicit differentiation, we take nothing for granted. we always identify what we are taking the derivative of, and with respect to what. That is, if we differentiate a y-term with respect to x, we attach the notation to it. This means we took the derivative of y with respect to x. We can simply call this y'. We don't attach anything if we differentiate a variable that is the same as the variable that we are differentiating with respect to. That is, if we differentiate x with respect to x, we don't attach anything. Why is that? Technically we do, if we differentiate an x-term with respect to x, we attach , but since derivative notations can function as fractions, we simplify this to 1, so you don't see it. If we differentiate a term with both x's and y's in it, we use the product rule. and we attach the appropriate notation for each part. when we differentiate the x-term we attach nothing. when we differentiate the y-term, we attach
Let's see how this works
We proceed by Implicit differentiation
.......remember, the derivative of a constant is zero, so the 16 disappears. we attach y' to the derivative of the y-term, but nothing to the derivative of the x-term as explained above.
Now we simply solve for y'
Now try the second derivative