That's ok. This implicit differentiation stuff can be kind of confusing at first. Note that is a function of x, as in . Now when we differentiate , we do so by differentiating with respect to x. So we note that is analagous to .
Now, when we differentiate functions that contain terms, we need to apply the chain rule, since after all, is a function of x. For example, when we differentiate , we know that by chain rule, this is the same as . Now above, we mentioned that is the same as . So it follows then that .
Try to apply this idea to other functions of .
Does this clarify things?
In an sense, yes. What you really end up multiplying by is the derivative of the exponent. Its more of a chain rule application: .And the second one how did we end up with 50? Is it bringing the exponent down?
So when I differentiated , it turned out to be
Does this demystify things?